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== 10.4#h(0.6em)Use the Properties of Logarithms

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Evaluate: ⓐ #math.equation(block: false, alt: "a to the power 0")[$a^(0)$] ⓑ #math.equation(block: false, alt: "a to the power 1 .")[$a^(1) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
ⓐ #math.equation(block: false, alt: "1")[$1$]; ⓑ #math.equation(block: false, alt: "a")[$a$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write with a rational exponent: #math.equation(block: false, alt: "the cube root of x squared y .")[$root(3, x^(2) y) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x squared y to the power the fraction 1 over 3")[$attach(x^(2) y, t: frac(1, 3))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Round to three decimal places: 2.5646415. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2.565")[$2.565$]
]
]


=== Use the Properties of Logarithms

Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.

The first two properties derive from the definition of logarithms. Since #math.equation(block: false, alt: "a to the power 0 equals 1 ,")[$a^(0) = 1 ,$] we can convert this to logarithmic form and get #math.equation(block: false, alt: "log sub a 1 equals 0 .")[$"log"_(a) 1 = 0 .$] Also, since #math.equation(block: false, alt: "a to the power 1 equals a ,")[$a^(1) = a ,$] we get #math.equation(block: false, alt: "log sub a a equals 1 .")[$"log"_(a) a = 1 .$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Logarithms]

#math.equation(block: true, alt: "log sub a 1 equals 0 log sub a a equals 1")[$"log"_(a) 1 = 0 #h(5em) "log"_(a) a = 1$]
]

In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but  recognizing and then applying the properties saves time.

#examplebox("Example 1")[][
Evaluate using the properties of logarithms: ⓐ #math.equation(block: false, alt: "log sub 8 1")[$"log"_(8) 1$] and ⓑ #math.equation(block: false, alt: "log sub 6 6 .")[$"log"_(6) 6 .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 8 1")[$#h(2em) "log"_(8) 1$]]),
  [Use the property, #math.equation(block: false, alt: "log sub a 1 equals 0")[$"log"_(a) 1 = 0$].], [#math.equation(block: false, alt: "0 log sub 8 1 equals 0")[$#h(3em) 0 #h(2em) "log"_(8) 1 = 0$]],
))

ⓑ  #linebreak() #math.equation(block: false, alt: "log sub 6 6; Use the property, log sub a a equals 1 ., 1, log sub 6 6 equals 1")[$&  &  & #h(2em) "log"_(6) 6 &  &  \ "Use the property," #h(0.2em) "log"_(a) a = 1 . &  &  & #h(3em) 1 &  & #h(2em) "log"_(6) 6 = 1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate using the properties of logarithms: ⓐ #math.equation(block: false, alt: "log sub 13 1")[$"log"_(13) 1$] ⓑ #math.equation(block: false, alt: "log sub 9 9 .")[$"log"_(9) 9 .$]

#solutionbox[
ⓐ 0 ⓑ 1
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate using the properties of logarithms: ⓐ #math.equation(block: false, alt: "log sub 5 1")[$"log"_(5) 1$] ⓑ #math.equation(block: false, alt: "log sub 7 7 .")[$"log"_(7) 7 .$]

#solutionbox[
ⓐ 0 ⓑ 1
]
]

The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.

The exponential equation #math.equation(block: false, alt: "a to the power log sub a x equals x")[$a^("log"_(a) x) = x$] converts to the logarithmic equation #math.equation(block: false, alt: "log sub a x equals log sub a x ,")[$"log"_(a) x = "log"_(a) x ,$] which is a true statement for positive values for #emph[x] only.

The logarithmic equation #math.equation(block: false, alt: "log sub a a to the power x equals x")[$"log"_(a) a^(x) = x$] converts to the exponential equation #math.equation(block: false, alt: "a to the power x equals a to the power x ,")[$a^(x) = a^(x) ,$] which is also a true statement.

These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the #strong[composition] of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Inverse Properties of Logarithms]

For #math.equation(block: false, alt: "a greater than 0 ,")[$a > 0 ,$]#math.equation(block: false, alt: "x greater than 0")[$x > 0$] and #math.equation(block: false, alt: "a not equal to 1 ,")[$a ≠ 1 ,$]

#math.equation(block: true, alt: "a to the power log sub a x equals x log sub a a to the power x equals x")[$a^("log"_(a) x) = x #h(5em) "log"_(a) a^(x) = x$]
]

In the next example, apply the inverse properties of logarithms.

#examplebox("Example 2")[][
Evaluate using the properties of logarithms: ⓐ #math.equation(block: false, alt: "4 to the power log sub 4 9")[$4^("log"_(4) 9)$] and ⓑ #math.equation(block: false, alt: "log sub 3 3 to the power 5 .")[$"log"_(3) 3^(5) .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 to the power log sub 4 9")[$#h(2em) 4^("log"_(4) 9)$]]),
  [Use the property, #math.equation(block: false, alt: "a to the power log sub a x equals x")[$a^("log"_(a) x) = x$].], [#math.equation(block: false, alt: "9 4 to the power log sub 4 9 equals 9")[$#h(3em) 9 #h(2em) 4^("log"_(4) 9) = 9$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 3 3 to the power 5")[$#h(2em) "log"_(3) 3^(5)$]]),
  [Use the property, #math.equation(block: false, alt: "a to the power log sub a x equals x")[$a^("log"_(a) x) = x$].], [#math.equation(block: false, alt: "5 log sub 3 3 to the power 5 equals 5")[$#h(3em) 5 #h(2em) "log"_(3) 3^(5) = 5$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate using the properties of logarithms: ⓐ #math.equation(block: false, alt: "5 to the power log sub 5 15")[$5^("log"_(5) 15)$] ⓑ #math.equation(block: false, alt: "log sub 7 7 to the power 4 .")[$"log"_(7) 7^(4) .$]

#solutionbox[
ⓐ 15 ⓑ 4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate using the properties of logarithms: ⓐ #math.equation(block: false, alt: "2 to the power log sub 2 8")[$2^("log"_(2) 8)$] ⓑ #math.equation(block: false, alt: "log sub 2 2 to the power 15 .")[$"log"_(2) 2^(15) .$]

#solutionbox[
ⓐ 8 ⓑ 15
]
]

There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.

In the Product Property of Exponents, #math.equation(block: false, alt: "a to the power m times a to the power n equals a to the power m plus n ,")[$a^(m) · a^(n) = a^(m + n) ,$] we see that to multiply the same base, we add the exponents. The #strong[Product Property of Logarithms], #math.equation(block: false, alt: "log sub a M times N equals log sub a M plus log sub a N")[$"log"_(a) M · N = "log"_(a) M + "log"_(a) N$] tells us to take the log of a product, we add the log of the factors.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Product Property of Logarithms]

If #math.equation(block: false, alt: "M greater than 0 , N greater than 0 , a greater than 0")[$M > 0 , N > 0 "," #h(0.2em) "a" > 0$] and #math.equation(block: false, alt: "a not equal to 1 ,")[$"a" ≠ 1 ,$] then,

#math.equation(block: true, alt: "log sub a open parenthesis M times N close parenthesis equals log sub a M plus log sub a N")[$"log"_(a) ( M · N ) = "log"_(a) M + "log"_(a) N$]The logarithm of a product is the sum of the logarithms.
]

We use this property to write the log of a product as a sum of the logs of each factor.

#examplebox("Example 3")[][
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible: ⓐ #math.equation(block: false, alt: "log sub 3 7 x")[$"log"_(3) 7 x$] and ⓑ #math.equation(block: false, alt: "log sub 4 64 x y .")[$"log"_(4) 64 x y .$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 3 7 x")[$#h(2em) "log"_(3) 7 x$]]),
  [Use the Product Property, #math.equation(block: false, alt: "log sub a open parenthesis M times N close parenthesis equals log sub a M plus log sub a N")[$"log"_(a) ( M · N ) = "log"_(a) M + "log"_(a) N$].], [#math.equation(block: false, alt: "log sub 3 7 plus log sub 3 x")[$#h(2em) "log"_(3) 7 + "log"_(3) x$]],
  [], [#math.equation(block: false, alt: "log sub 3 7 x equals log sub 3 7 plus log sub 3 x")[$#h(2em) "log"_(3) 7 x = "log"_(3) 7 + "log"_(3) x$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 4 64 x y")[$#h(2em) "log"_(4) 64 x y$]]),
  [Use the Product Property, #math.equation(block: false, alt: "log sub a open parenthesis M times N close parenthesis equals log sub a M plus log sub a N")[$"log"_(a) ( M · N ) = "log"_(a) M + "log"_(a) N$].], [#math.equation(block: false, alt: "log sub 4 64 plus log sub 4 x plus log sub 4 y")[$#h(2em) "log"_(4) 64 + "log"_(4) x + "log"_(4) y$]],
  [Simplify by evaluating #math.equation(block: false, alt: "log sub 4 64")[$"log"_(4) 64$].], [#math.equation(block: false, alt: "3 plus log sub 4 x plus log sub 4 y")[$#h(2em) 3 + log_(4) x + log_(4) y$]],
  [], [#math.equation(block: false, alt: "log sub 4 64 x y equals 3 plus log sub 4 x plus log sub 4 y")[$#h(2em) "log"_(4) 64 x y = 3 + "log"_(4) x + "log"_(4) y$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ #math.equation(block: false, alt: "log sub 3 3 x")[$"log"_(3) 3 x$] ⓑ #math.equation(block: false, alt: "log sub 2 8 x y")[$"log"_(2) 8 x y$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "1 plus log sub 3 x")[$1 + "log"_(3) x$] #linebreak() ⓑ #math.equation(block: false, alt: "3 plus log sub 2 x plus log sub 2 y")[$3 + "log"_(2) x + "log"_(2) y$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ #math.equation(block: false, alt: "log sub 9 9 x")[$"log"_(9) 9 x$] ⓑ #math.equation(block: false, alt: "log sub 3 27 x y")[$"log"_(3) 27 x y$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "1 plus log sub 9 x")[$1 + "log"_(9) x$] #linebreak() ⓑ #math.equation(block: false, alt: "3 plus log sub 3 x plus log sub 3 y")[$3 + "log"_(3) x + "log"_(3) y$]
]
]

Similarly, in the Quotient Property of Exponents, #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n ,")[$frac(a^(m), a^(n)) = a^(m − n) ,$] we see that to divide the same base, we subtract the exponents. The #strong[Quotient Property of Logarithms], #math.equation(block: false, alt: "log sub a the fraction M over N equals log sub a M minus log sub a N")[$"log"_(a) frac(M, N) = "log"_(a) M − "log"_(a) N$] tells us to take the log of a quotient, we subtract the log of the numerator and denominator.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Quotient Property of Logarithms]

If #math.equation(block: false, alt: "M greater than 0 , N greater than 0 , a greater than 0")[$M > 0 , N > 0 "," "a" > 0 #h(0.2em)$] and #math.equation(block: false, alt: "a not equal to 1 ,")[$"a" ≠ 1 ,$] then,

#math.equation(block: true, alt: "log sub a the fraction M over N equals log sub a M minus log sub a N")[$"log"_(a) frac(M, N) = "log"_(a) M − "log"_(a) N$]The logarithm of a quotient is the difference of the logarithms.
]

Note that #math.equation(block: false, alt: "log sub a M minus log sub a N not equal to log sub a open parenthesis M minus N close parenthesis .")[$"log"_(a) M − "log"_(a) N ≠ "log"_(a) ( M − N ) .$]

We use this property to write the log of a quotient as a difference of the logs of each factor.

#examplebox("Example 4")[][
Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible. #linebreak() ⓐ #math.equation(block: false, alt: "log sub 5 the fraction 5 over 7")[$"log"_(5) frac(5, 7)$] and ⓑ #math.equation(block: false, alt: "log the fraction x over 100")[$"log" frac(x, 100)$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 5 the fraction 5 over 7")[$#h(2em) "log"_(5) frac(5, 7)$]]),
  [Use the Quotient Property, #math.equation(block: false, alt: "log sub a the fraction M over N equals log sub a M minus log sub a N")[$"log"_(a) frac(M, N) = "log"_(a) M − "log"_(a) N$].], [#math.equation(block: false, alt: "log sub 5 5 minus log sub 5 7")[$#h(2em) "log"_(5) 5 − "log"_(5) 7$]],
  [Simplify.], [#math.equation(block: false, alt: "1 minus log sub 5 7")[$#h(2em) 1 − "log"_(5) 7$]],
  [], [#math.equation(block: false, alt: "log sub 5 the fraction 5 over 7 equals 1 minus log sub 5 7")[$#h(2em) "log"_(5) frac(5, 7) = 1 − "log"_(5) 7$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log the fraction x over 100")[$#h(2em) "log" frac(x, 100)$]]),
  [Use the Quotient Property, #math.equation(block: false, alt: "log sub a the fraction M over N equals log sub a M minus log sub a N")[$"log"_(a) frac(M, N) = "log"_(a) M − "log"_(a) N$].], [#math.equation(block: false, alt: "log x minus log 100")[$#h(2em) "log" #h(0.2em) x − "log" 100$]],
  [Simplify.], [#math.equation(block: false, alt: "log x minus 2")[$#h(2em) "log" #h(0.2em) x − 2$]],
  [], [#math.equation(block: false, alt: "log the fraction x over 100 equals log x minus 2")[$#h(2em) "log" frac(x, 100) = "log" #h(0.2em) x − 2$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ #math.equation(block: false, alt: "log sub 4 the fraction 3 over 4")[$"log"_(4) frac(3, 4)$] ⓑ #math.equation(block: false, alt: "log the fraction x over 1000")[$"log" frac(x, 1000)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "log sub 4 3 minus 1")[$"log"_(4) 3 − 1$] ⓑ #math.equation(block: false, alt: "log x minus 3")[$"log" #h(0.2em) x − 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ #math.equation(block: false, alt: "log sub 2 the fraction 5 over 4")[$"log"_(2) frac(5, 4)$] ⓑ #math.equation(block: false, alt: "log the fraction 10 over y")[$"log" frac(10, y)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "log sub 2 5 minus 2")[$"log"_(2) 5 − 2$] ⓑ #math.equation(block: false, alt: "1 minus log y")[$1 − "log" y$]
]
]

The third property of logarithms is related to the Power Property of Exponents, #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n ,")[$attach(( a^(m) ), t: n) = a^(m · n) ,$] we see that to raise a power to a power, we multiply the exponents. The #strong[Power Property of Logarithms], #math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$] tells us to take the log of a number raised to a power, we multiply the power times the log of the number.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Power Property of Logarithms]

If #math.equation(block: false, alt: "M greater than 0 , a greater than 0 , a not equal to 1")[$M > 0 , #h(0.2em) "a" > 0 , #h(0.2em) "a" ≠ 1$] and #math.equation(block: false, alt: "p")[$p$] is any real number then,

#math.equation(block: true, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$]The log of a number raised to a power is the product of the power times the log of the number.
]

We use this property to write the log of a number raised to a power as the product of the power times the log of the number. We essentially take the exponent and throw it in front of the logarithm.

#examplebox("Example 5")[][
Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible. #linebreak() ⓐ #math.equation(block: false, alt: "log sub 5 4 cubed")[$"log"_(5) 4^(3)$] and ⓑ #math.equation(block: false, alt: "log x to the power 10")[$"log" x^(10)$]

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 5 4 cubed")[$#h(2em) "log"_(5) 4^(3)$]]),
  [Use the Power Property, #math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$].], [#math.equation(block: false, alt: "3 log sub 5 4")[$#h(2em) 3 "log"_(5) 4$]],
  [], [#math.equation(block: false, alt: "log sub 5 4 cubed equals 3 log sub 5 4")[$#h(2em) "log"_(5) 4^(3) = 3 "log"_(5) 4$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log x to the power 10")[$#h(2em) "log" x^(10)$]]),
  [Use the Power Property, #math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$].], [#math.equation(block: false, alt: "10 log x")[$#h(2em) 10 #h(0.2em) "log" #h(0.2em) x$]],
  [], [#math.equation(block: false, alt: "log x to the power 10 equals 10 log x")[$#h(2em) "log" x^(10) = 10 "log" #h(0.2em) x$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ #math.equation(block: false, alt: "log sub 7 5 to the power 4")[$"log"_(7) 5^(4)$] ⓑ #math.equation(block: false, alt: "log x to the power 100")[$"log" x^(100)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "4 log sub 7 5")[$4 "log"_(7) 5$] ⓑ #math.equation(block: false, alt: "100 times log x")[$100 · "log" #h(0.2em) x$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ #math.equation(block: false, alt: "log sub 2 3 to the power 7")[$"log"_(2) 3^(7)$] ⓑ #math.equation(block: false, alt: "log x to the power 20")[$"log" x^(20)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "7 log sub 2 3")[$7 "log"_(2) 3$] ⓑ #math.equation(block: false, alt: "20 times log x")[$20 · "log" #h(0.2em) x$]
]
]

We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Logarithms]

If #math.equation(block: false, alt: "M greater than 0 , N greater than 0 , a greater than 0 , a not equal to 1")[$M > 0 , #h(0.2em) N > 0 , "a" > 0 , #h(0.2em) "a" ≠ 1$] and #math.equation(block: false, alt: "p")[$p$] is any real number then,

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Property], [Base #math.equation(block: false, alt: "a")[$a$]], [Base #math.equation(block: false, alt: "e")[$e$]]),
  [], [#math.equation(block: false, alt: "log sub a 1 equals 0")[$"log"_(a) 1 = 0$]], [#math.equation(block: false, alt: "ln 1 equals 0")[$"ln" #h(0.2em) 1 = 0$]],
  [], [#math.equation(block: false, alt: "log sub a a equals 1")[$"log"_(a) a = 1$]], [#math.equation(block: false, alt: "ln e equals 1")[$"ln" #h(0.2em) e = 1$]],
  [#strong[Inverse Properties]], [#math.equation(block: false, alt: "a to the power log sub a x equals x; log sub a a to the power x equals x")[$a^("log"_(a) x) = x \ "log"_(a) a^(x) = x$]], [#math.equation(block: false, alt: "e to the power ln x equals x; ln e to the power x equals x")[$e^("ln" #h(0.2em) x) = x \ "ln" #h(0.2em) e^(x) = x$]],
  [#strong[Product Property of Logarithms]], [#math.equation(block: false, alt: "log sub a open parenthesis M times N close parenthesis equals log sub a M plus log sub a N")[$"log"_(a) ( M · N ) = "log"_(a) M + "log"_(a) N$]], [#math.equation(block: false, alt: "ln open parenthesis M times N close parenthesis equals ln M plus ln N")[$"ln" ( M #h(0.2em) · #h(0.2em) N ) = "ln" #h(0.2em) M + "ln" #h(0.2em) N$]],
  [#strong[Quotient Property of Logarithms]], [#math.equation(block: false, alt: "log sub a the fraction M over N equals log sub a M minus log sub a N")[$"log"_(a) frac(M, N) = "log"_(a) M − "log"_(a) N$]], [#math.equation(block: false, alt: "ln the fraction M over N equals ln M minus ln N")[$"ln" #h(0.2em) frac(M, N) = "ln" #h(0.2em) M − "ln" #h(0.2em) N$]],
  [#strong[Power Property of Logarithms]], [#math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$]], [#math.equation(block: false, alt: "ln M to the power p equals p ln M")[$"ln" #h(0.2em) M^(p) = p #h(0.2em) "ln" #h(0.2em) M$]],
))
]

Now that we have the properties we can use them to “expand” a logarithmic expression. This means to write the logarithm as a sum or difference and without any powers.

We generally apply the Product and Quotient Properties before we apply the Power Property.

#examplebox("Example 6")[][
Use the Properties of Logarithms to expand the logarithm #math.equation(block: false, alt: "log sub 4 open parenthesis 2 x cubed y squared close parenthesis")[$"log"_(4) ( 2 x^(3) y^(2) )$]. Simplify, if possible.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 4 open parenthesis 2 x cubed y squared close parenthesis")[$#h(2em) "log"_(4) ( 2 x^(3) y^(2) )$]]),
  [Use the Product Property, #math.equation(block: false, alt: "log sub a M times N equals log sub a M plus log sub a N")[$"log"_(a) M · N = "log"_(a) M + "log"_(a) N$].], [#math.equation(block: false, alt: "log sub 4 2 plus log sub 4 x cubed plus log sub 4 y squared")[$#h(2em) "log"_(4) 2 + "log"_(4) x^(3) + "log"_(4) y^(2)$]],
  [Use the Power Property, #math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$], on the last two terms.], [#math.equation(block: false, alt: "log sub 4 2 plus 3 log sub 4 x plus 2 log sub 4 y")[$#h(2em) "log"_(4) 2 + 3 "log"_(4) x + 2 "log"_(4) y$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 2 plus 3 log sub 4 x plus 2 log sub 4 y")[$#h(2em) frac(1, 2) + 3 "log"_(4) x + 2 "log"_(4) y$]],
  [], [#math.equation(block: false, alt: "log sub 4 open parenthesis 2 x cubed y squared close parenthesis equals the fraction 1 over 2 plus 3 log sub 4 x plus 2 log sub 4 y")[$#h(2em) "log"_(4) ( 2 x^(3) y^(2) ) = frac(1, 2) + 3 "log"_(4) x + 2 "log"_(4) y$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to expand the logarithm #math.equation(block: false, alt: "log sub 2 open parenthesis 5 x to the power 4 y squared close parenthesis")[$"log"_(2) ( 5 x^(4) y^(2) )$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "log sub 2 5 plus 4 log sub 2 x plus 2 log sub 2 y")[$"log"_(2) 5 + 4 "log"_(2) x + 2 "log"_(2) y$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to expand the logarithm #math.equation(block: false, alt: "log sub 3 open parenthesis 7 x to the power 5 y cubed close parenthesis")[$"log"_(3) ( 7 x^(5) y^(3) )$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "log sub 3 7 plus 5 log sub 3 x plus 3 log sub 3 y")[$"log"_(3) 7 + 5 "log"_(3) x + 3 "log"_(3) y$]
]
]

When we have a radical in the logarithmic expression, it is helpful to first write its radicand as a rational exponent.

#examplebox("Example 7")[][
Use the Properties of Logarithms to expand the logarithm #math.equation(block: false, alt: "log sub 2 the 4th root of the fraction x cubed over 3 y squared z")[$"log"_(2) root(4, frac(x^(3), 3 y^(2) z))$]. Simplify, if possible.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "log sub 2 the 4th root of the fraction x cubed over 3 y squared z")[$#h(2em) "log"_(2) root(4, frac(x^(3), 3 y^(2) z))$]]),
  [Rewrite the radical with a rational exponent.], [#math.equation(block: false, alt: "log sub 2 open parenthesis the fraction x cubed over 3 y squared z close parenthesis to the power the fraction 1 over 4")[$#h(2em) "log"_(2) attach(( frac(x^(3), 3 y^(2) z) ), t: frac(1, 4))$]],
  [Use the Power Property, #math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$].], [#math.equation(block: false, alt: "the fraction 1 over 4 log sub 2 open parenthesis the fraction x cubed over 3 y squared z close parenthesis")[$#h(2em) frac(1, 4) "log"_(2) ( frac(x^(3), 3 y^(2) z) )$]],
  [Use the Quotient Property, #math.equation(block: false, alt: "log sub a M times N equals log sub a M minus log sub a N")[$"log"_(a) M · N = "log"_(a) M − "log"_(a) N$].], [#math.equation(block: false, alt: "the fraction 1 over 4 open parenthesis log sub 2 open parenthesis x cubed close parenthesis minus log sub 2 open parenthesis 3 y squared z close parenthesis close parenthesis")[$#h(2em) frac(1, 4) ( "log"_(2) ( x^(3) ) − "log"_(2) ( 3 y^(2) z ) )$]],
  [Use the Product Property, #math.equation(block: false, alt: "log sub a M times N equals log sub a M plus log sub a N")[$"log"_(a) M · N = "log"_(a) M + "log"_(a) N$], in the second term.], [#math.equation(block: false, alt: "the fraction 1 over 4 open parenthesis log sub 2 open parenthesis x cubed close parenthesis minus open parenthesis log sub 2 3 plus log sub 2 y squared plus log sub 2 z close parenthesis close parenthesis")[$#h(2em) frac(1, 4) ( "log"_(2) ( x^(3) ) − ( "log"_(2) 3 + "log"_(2) y^(2) + "log"_(2) z ) )$]],
  [Use the Power Property, #math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$], inside the parentheses.], [#math.equation(block: false, alt: "the fraction 1 over 4 open parenthesis 3 log sub 2 x minus open parenthesis log sub 2 3 plus 2 log sub 2 y plus log sub 2 z close parenthesis close parenthesis")[$#h(2em) frac(1, 4) ( 3 "log"_(2) x − ( "log"_(2) 3 + 2 "log"_(2) y + "log"_(2) z ) )$]],
  [Simplify by distributing.], [#math.equation(block: false, alt: "the fraction 1 over 4 open parenthesis 3 log sub 2 x minus log sub 2 3 minus 2 log sub 2 y minus log sub 2 z close parenthesis")[$#h(2em) frac(1, 4) ( 3 "log"_(2) x − "log"_(2) 3 − 2 "log"_(2) y − "log"_(2) z )$]],
  [], [#math.equation(block: false, alt: "log sub 2 the 4th root of the fraction x cubed over 3 y squared z equals the fraction 1 over 4 open parenthesis 3 log sub 2 x minus log sub 2 3 minus 2 log sub 2 y minus log sub 2 z close parenthesis")[$"log"_(2) root(4, frac(x^(3), 3 y^(2) z)) = frac(1, 4) ( 3 "log"_(2) x − "log"_(2) 3 − 2 "log"_(2) y − "log"_(2) z )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to expand the logarithm #math.equation(block: false, alt: "log sub 4 the 5th root of the fraction x to the power 4 over 2 y cubed z squared")[$"log"_(4) root(5, frac(x^(4), 2 y^(3) z^(2)))$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 5 open parenthesis 4 log sub 4 x minus the fraction 1 over 2 minus 3 log sub 4 y minus 2 log sub 4 z close parenthesis")[$frac(1, 5) ( 4 "log"_(4) x − frac(1, 2) − 3 "log"_(4) y − 2 "log"_(4) z )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to expand the logarithm #math.equation(block: false, alt: "log sub 3 the cube root of the fraction x squared over 5 y to the power z")[$"log"_(3) root(3, frac(x^(2), 5 y^ z))$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis 2 log sub 3 x minus log sub 3 5 minus log sub 3 y minus log sub 3 z close parenthesis")[$frac(1, 3) ( 2 "log"_(3) x − "log"_(3) 5 − "log"_(3) y − "log"_(3) z )$]
]
]

The opposite of expanding a logarithm is to condense a sum or difference of logarithms that have the same base into a single logarithm. We again use the properties of logarithms to help us, but in reverse.

To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.

#examplebox("Example 8")[][
Use the Properties of Logarithms to condense the logarithm #math.equation(block: false, alt: "log sub 4 3 plus log sub 4 x minus log sub 4 y")[$"log"_(4) 3 + "log"_(4) x − "log"_(4) y$]. Simplify, if possible.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([The log expressions all have the same base, 4.], [#math.equation(block: false, alt: "log sub 4 3 plus log sub 4 x minus log sub 4 y")[$#h(6em) "log"_(4) 3 + "log"_(4) x − "log"_(4) y$]]),
  [The first two terms are added, so we use the Product Property, #math.equation(block: false, alt: "log sub a M plus log sub a N equals log sub a M times N")[$"log"_(a) M + "log"_(a) N = "log"_(a) M · N$].], [#math.equation(block: false, alt: "log sub 4 3 x minus log sub 4 y")[$#h(6em) "log"_(4) 3 x − "log"_(4) y$]],
  [Since the logs are subtracted, we use the Quotient Property, #math.equation(block: false, alt: "log sub a M minus log sub a N equals log sub a the fraction M over N")[$"log"_(a) M − "log"_(a) N = "log"_(a) frac(M, N)$].], [#math.equation(block: false, alt: "log sub 4 the fraction 3 x over y")[$#h(6em) "log"_(4) frac(3 x, y)$]],
  [], [#math.equation(block: false, alt: "log sub 4 3 plus log sub 4 x minus log sub 4 y equals log sub 4 the fraction 3 x over y")[$#h(6em) "log"_(4) 3 + "log"_(4) x − "log"_(4) y = "log"_(4) frac(3 x, y)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to condense the logarithm #math.equation(block: false, alt: "log sub 2 5 plus log sub 2 x minus log sub 2 y")[$"log"_(2) 5 + "log"_(2) x − "log"_(2) y$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "log sub 2 the fraction 5 x over y")[$"log"_(2) frac(5 x, y)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to condense the logarithm #math.equation(block: false, alt: "log sub 3 6 minus log sub 3 x minus log sub 3 y")[$"log"_(3) 6 − "log"_(3) x − "log"_(3) y$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "log sub 3 the fraction 6 over x y")[$"log"_(3) frac(6, x y)$]
]
]

#examplebox("Example 9")[][
Use the Properties of Logarithms to condense the logarithm #math.equation(block: false, alt: "2 log sub 3 x plus 4 log sub 3 open parenthesis x plus 1 close parenthesis")[$2 "log"_(3) x + 4 "log"_(3) ( x + 1 )$]. Simplify, if possible.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([The log expressions have the same base, 3.], [#math.equation(block: false, alt: "2 log sub 3 x plus 4 log sub 3 open parenthesis x plus 1 close parenthesis")[$#h(2em) 2 "log"_(3) x + 4 "log"_(3) ( x + 1 )$]]),
  [Use the Power Property, #math.equation(block: false, alt: "log sub a M plus log sub a N equals log sub a M times N")[$"log"_(a) M + "log"_(a) N = "log"_(a) M · N$].], [#math.equation(block: false, alt: "log sub 3 x squared plus log sub 3 open parenthesis x plus 1 close parenthesis to the power 4")[$#h(2em) "log"_(3) x^(2) + "log"_(3) attach(( x + 1 ), t: 4)$]],
  [The terms are added, so we use the Product Property, #math.equation(block: false, alt: "log sub a M plus log sub a N equals log sub a M times N")[$"log"_(a) M + "log"_(a) N = "log"_(a) M · N$].], [#math.equation(block: false, alt: "log sub 3 x squared open parenthesis x plus 1 close parenthesis to the power 4")[$#h(2em) "log"_(3) x^(2) attach(( x + 1 ), t: 4)$]],
  [], [#math.equation(block: false, alt: "2 log sub 3 x plus 4 log sub 3 open parenthesis x plus 1 close parenthesis equals log sub 3 x squared open parenthesis x plus 1 close parenthesis to the power 4")[$#h(2em) 2 "log"_(3) x + 4 "log"_(3) ( x + 1 ) = "log"_(3) x^(2) attach(( x + 1 ), t: 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to condense the logarithm #math.equation(block: false, alt: "3 log sub 2 x plus 2 log sub 2 open parenthesis x minus 1 close parenthesis")[$3 "log"_(2) x + 2 "log"_(2) ( x − 1 )$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "log sub 2 x cubed open parenthesis x minus 1 close parenthesis squared")[$"log"_(2) x^(3) attach(( x − 1 ), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Properties of Logarithms to condense the logarithm #math.equation(block: false, alt: "2 log x plus 2 log open parenthesis x plus 1 close parenthesis")[$2 "log" #h(0.2em) x + 2 "log" ( x + 1 )$]. Simplify, if possible.

#solutionbox[
#math.equation(block: true, alt: "log x squared open parenthesis x plus 1 close parenthesis squared")[$"log" x^(2) attach(( x + 1 ), t: 2)$]
]
]


=== Use the Change-of-Base Formula

To evaluate a logarithm with any other base, we can use the #strong[Change-of-Base Formula]. We will show how this is derived.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Suppose we want to evaluate #math.equation(block: false, alt: "log sub a M")[$"log"_(a) M$].], [#math.equation(block: false, alt: "log sub a M")[$#h(2em) "log"_(a) M #h(4.5em)$]]),
  [Let #math.equation(block: false, alt: "y equals log sub a M")[$y = "log"_(a) M$].], [#math.equation(block: false, alt: "y equals log sub a M")[$#h(2em) y #h(0.2em) = #h(0.2em) "log"_(a) M$]],
  [Rewrite the expression in exponential form.], [#math.equation(block: false, alt: "a to the power y equals M")[$#h(2em) a^(y) #h(0.2em) = #h(0.2em) M #h(1.7em)$]],
  [Take the #math.equation(block: false, alt: "log sub b")[$"log"_(b)$] of each side.], [#math.equation(block: false, alt: "log sub b a to the power y equals log sub b M")[$#h(2em) "log"_(b) a^(y) #h(0.2em) = #h(0.2em) "log"_(b) M$]],
  [Use the Power Property.], [#math.equation(block: false, alt: "y log sub b a equals log sub b M")[$#h(2em) y "log"_(b) a #h(0.2em) = #h(0.2em) "log"_(b) M$]],
  [Solve for #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "y equals the fraction log sub b M over log sub b a")[$#h(2em) y #h(0.2em) = #h(0.2em) frac("log"_(b) M, "log"_(b) a)$]],
  [Substitute #math.equation(block: false, alt: "y equals log sub a M")[$y = "log"_(a) M$].], [#math.equation(block: false, alt: "log sub a M equals the fraction log sub b M over log sub b a")[$#h(2em) "log"_(a) M #h(0.2em) = #h(0.2em) frac("log"_(b) M, "log"_(b) a)$]],
))

The Change-of-Base Formula introduces a new base #math.equation(block: false, alt: "b .")[$b .$] This can be any base #emph[b] we want where #math.equation(block: false, alt: "b greater than 0 , b not equal to 1 .")[$b > 0 , b ≠ 1 .$] Because our calculators have keys for logarithms base 10 and base #emph[e], we will rewrite the Change-of-Base Formula with the new base as 10 or #emph[e].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Change-of-Base Formula]

For any logarithmic bases #math.equation(block: false, alt: "a , b")[$a , b$] and #math.equation(block: false, alt: "M greater than 0 ,")[$M > 0 ,$]

#math.equation(block: true, alt: "log sub a M equals the fraction log sub b M over log sub b a, log sub a M equals the fraction log M over log a, log sub a M equals the fraction ln M over ln a; new base b, new base 10, new base e")[$"log"_(a) M = frac("log"_(b) M, "log"_(b) a) &  &  & #h(2em) "log"_(a) M = frac("log" M, "log" a) &  &  & #h(2em) "log"_(a) M = frac("ln" #h(0.2em) M, "ln" #h(0.2em) a) \ "new base" #h(0.2em) b &  &  & #h(2em) "new base 10" &  &  & #h(2em) "new base" #h(0.2em) e$]
]

When we use a calculator to find the logarithm value, we usually round to three decimal places. This gives us an approximate value and so we use the approximately equal symbol #math.equation(block: false, alt: "(≈)")[$"(≈)"$].

#examplebox("Example 10")[][
Rounding to three decimal places, approximate #math.equation(block: false, alt: "log sub 4 35 .")[$"log"_(4) 35 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical expression log base 4 of 35 is displayed in black text on a white background.], alt: "The mathematical expression log base 4 of 35 is displayed in black text on a white background.", caption: none)]),
  [Use the Change-of-Base Formula.], [#figure(figph[The image displays the logarithm change of base formula: log subscript 'a' of M equals the fraction of log subscript 'b' of M over log subscript 'b' of 'a'.], alt: "The image displays the logarithm change of base formula: log subscript 'a' of M equals the fraction of log subscript 'b' of M over log subscript 'b' of 'a'.", caption: none)],
  [Identify #emph[a] and #emph[M]. Choose 10 for #emph[b].], [#figure(figph[The image displays the logarithm change of base formula, showing that log base 4 of 35 is equal to the ratio of log 35 to log 4, illustrating how to convert logarithms to a different base.], alt: "The image displays the logarithm change of base formula, showing that log base 4 of 35 is equal to the ratio of log 35 to log 4, illustrating how to convert logarithms to a different base.", caption: none)],
  [Enter the expression #math.equation(block: false, alt: "the fraction log 35 over log 4")[$frac("log" 35, "log" 4)$] in the calculator #linebreak() using the log button for base 10. Round to three decimal places.], [#figure(figph[The image displays the logarithm expression 'log base 4 of 35 is approximately equal to 2.565' in black text on a white background.], alt: "The image displays the logarithm expression 'log base 4 of 35 is approximately equal to 2.565' in black text on a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Rounding to three decimal places, approximate #math.equation(block: false, alt: "log sub 3 42 .")[$"log"_(3) 42 .$]

#solutionbox[
#math.equation(block: true, alt: "3.402")[$3.402$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Rounding to three decimal places, approximate #math.equation(block: false, alt: "log sub 5 46 .")[$"log"_(5) 46 .$]

#solutionbox[
#math.equation(block: true, alt: "2.379")[$2.379$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with using the properties of logarithms.

- #link("https://openstax.org/l/37Logproperties")[Using Properties of Logarithms to Expand Logs]
- #link("https://openstax.org/l/37Condenselogs")[Using Properties of Logarithms to Condense Logs]
- #link("https://openstax.org/l/37Changeofbase")[Change of Base]
]


=== Key Concepts

- #strong[Properties of Logarithms] #linebreak() 
#math.equation(block: true, alt: "log sub a 1 equals 0 log sub a a equals 1")[$#h(3em) "log"_(a) 1 = 0 #h(5em) "log"_(a) a = 1$]
- #strong[Inverse Properties of Logarithms]
- For #math.equation(block: false, alt: "a greater than 0 ,")[$a > 0 ,$]#math.equation(block: false, alt: "x greater than 0")[$x > 0$] and #math.equation(block: false, alt: "a not equal to 1")[$a ≠ 1$] #linebreak() 
#math.equation(block: true, alt: "a to the power log sub a x equals x log sub a a to the power x equals x")[$a^("log"_(a) x) = x #h(4.95em) "log"_(a) a^(x) = x$]
- #strong[Product Property of Logarithms] #linebreak() 
- If #math.equation(block: false, alt: "M greater than 0 , N greater than 0 , a greater than 0")[$M > 0 , N > 0 "," #h(0.2em) "a" > 0$] and #math.equation(block: false, alt: "a not equal to 1 ,")[$"a" ≠ 1 ,$] then, #linebreak() 
#math.equation(block: true, alt: "log sub a M times N equals log sub a M plus log sub a N")[$#h(2.08em) "log"_(a) M · N = "log"_(a) M + "log"_(a) N$] #linebreak() 
The logarithm of a product is the sum of the logarithms.
- #strong[Quotient Property of Logarithms] #linebreak() 
- If #math.equation(block: false, alt: "M greater than 0 , N greater than 0 , a greater than 0")[$M > 0 , N > 0 "," #h(0.2em) "a" > 0$] and #math.equation(block: false, alt: "a not equal to 1 ,")[$"a" ≠ 1 ,$] then, #linebreak() 
#math.equation(block: true, alt: "log sub a the fraction M over N equals log sub a M minus log sub a N")[$#h(3em) "log"_(a) frac(M, N) = "log"_(a) M − "log"_(a) N$] #linebreak() 
The logarithm of a quotient is the difference of the logarithms.
- #strong[Power Property of Logarithms] #linebreak() 
- If #math.equation(block: false, alt: "M greater than 0 , a greater than 0 , a not equal to 1")[$M > 0 , #h(0.2em) "a" > 0 , #h(0.2em) "a" ≠ 1$] and #math.equation(block: false, alt: "p")[$p$] is any real number then, #linebreak() 
#math.equation(block: true, alt: "log sub a M to the power p equals p log sub a M")[$"log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$] #linebreak() 
The log of a number raised to a power is the product of the power times the log of the number.
- #strong[Properties of Logarithms Summary] #linebreak() 
If #math.equation(block: false, alt: "M greater than 0 , a greater than 0 , a not equal to 1")[$M > 0 , #h(0.2em) "a" > 0 , #h(0.2em) "a" ≠ 1$] and #math.equation(block: false, alt: "p")[$p$] is any real number then, #linebreak() 
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Property], [Base #math.equation(block: false, alt: "a")[$a$]], [Base #math.equation(block: false, alt: "e")[$e$]]),
  [], [#math.equation(block: false, alt: "log sub a 1 equals 0")[$"log"_(a) 1 = 0$]], [#math.equation(block: false, alt: "ln 1 equals 0")[$"ln" 1 = 0$]],
  [], [#math.equation(block: false, alt: "log sub a a equals 1")[$"log"_(a) a = 1$]], [#math.equation(block: false, alt: "ln e equals 1")[$"ln" #h(0.2em) e = 1$]],
  [#strong[Inverse Properties]], [#math.equation(block: false, alt: "a to the power log sub a x equals x; log sub a a to the power x equals x")[$#h(1.25em) a^("log"_(a) x) = x \ #h(1em) "log"_(a) a^(x) = x$]], [#math.equation(block: false, alt: "e to the power ln x equals x; ln e to the power x equals x")[$#h(1.2em) e^("ln" #h(0.2em) x) = x \ #h(1em) "ln" #h(0.2em) e^(x) = x$]],
  [#strong[Product Property of Logarithms]], [#math.equation(block: false, alt: "log sub a open parenthesis M times N close parenthesis equals log sub a M plus log sub a N")[$"log"_(a) ( M · N ) = "log"_(a) M + "log"_(a) N$]], [#math.equation(block: false, alt: "ln open parenthesis M times N close parenthesis equals ln M plus ln N")[$"ln" ( M #h(0.2em) · #h(0.2em) N ) = "ln" #h(0.2em) M + "ln" #h(0.2em) N$]],
  [#strong[Quotient Property of Logarithms]], [#math.equation(block: false, alt: "log sub a the fraction M over N equals log sub a M minus log sub a N")[$#h(1.9em) "log"_(a) frac(M, N) = "log"_(a) M − "log"_(a) N$]], [#math.equation(block: false, alt: "ln the fraction M over N equals ln M minus ln N")[$#h(2.4em) "ln" frac(M, N) = "ln" #h(0.2em) M − "ln" #h(0.2em) N$]],
  [#strong[Power Property of Logarithms]], [#math.equation(block: false, alt: "log sub a M to the power p equals p log sub a M")[$#h(1.55em) "log"_(a) M^(p) = p #h(0.2em) "log"_(a) M$]], [#math.equation(block: false, alt: "ln M to the power p equals p ln M")[$#h(1.76em) "ln" #h(0.2em) M^(p) = p "ln" #h(0.2em) M$]],
))
- #strong[Change-of-Base Formula] #linebreak() 
For any logarithmic bases #emph[a] and #emph[b], and #math.equation(block: false, alt: "M greater than 0 ,")[$M > 0 ,$] #linebreak() 
#math.equation(block: true, alt: "log sub a M equals the fraction log sub b M over log sub b a, log sub a M equals the fraction log M over log a, log sub a M equals the fraction ln M over ln a; new base b, new base 10, new base e")[$"log"_(a) M = frac("log"_(b) M, "log"_(b) a) &  &  & #h(2em) "log"_(a) M = frac("log" M, "log" a) &  &  & #h(2em) "log"_(a) M = frac("ln" #h(0.2em) M, "ln" #h(0.2em) a) \ "new base" #h(0.2em) b &  &  & #h(2em) "new base 10" &  &  & #h(2em) "new base" #h(0.2em) e$]


==== Practice Makes Perfect

#strong[Use the Properties of Logarithms]

In the following exercises, use the properties of logarithms to evaluate.

ⓐ #math.equation(block: false, alt: "log sub 4 1")[$"log"_(4) 1$] ⓑ #math.equation(block: false, alt: "log sub 8 8")[$"log"_(8) 8$]

ⓐ #math.equation(block: false, alt: "log sub 12 1")[$"log"_(12) 1$] ⓑ #math.equation(block: false, alt: "ln e")[$"ln" #h(0.2em) e$]

#solutionbox[
ⓐ 0 ⓑ 1
]

ⓐ #math.equation(block: false, alt: "3 to the power log sub 3 6")[$3^("log"_(3) 6)$] ⓑ #math.equation(block: false, alt: "log sub 2 2 to the power 7")[$"log"_(2) 2^(7)$]

ⓐ #math.equation(block: false, alt: "5 to the power log sub 5 10")[$5^("log"_(5) 10)$] ⓑ #math.equation(block: false, alt: "log sub 4 4 to the power 10")[$"log"_(4) 4^(10)$]

#solutionbox[
ⓐ 10 ⓑ 10
]

ⓐ #math.equation(block: false, alt: "8 to the power log sub 8 7")[$8^("log"_(8) 7)$] ⓑ #math.equation(block: false, alt: "log sub 6 6 to the power −2")[$"log"_(6) 6^(−2)$]

ⓐ #math.equation(block: false, alt: "6 to the power log sub 6 15")[$6^("log"_(6) 15)$] ⓑ #math.equation(block: false, alt: "log sub 8 8 to the power −4")[$"log"_(8) 8^(−4)$]

#solutionbox[
ⓐ 15 ⓑ #math.equation(block: false, alt: "−4")[$−4$]
]

ⓐ #math.equation(block: false, alt: "10 to the power log the square root of 5")[$10^("log" sqrt(5))$] ⓑ #math.equation(block: false, alt: "log 10 to the power −2")[$"log" 10^(−2)$]

ⓐ #math.equation(block: false, alt: "10 to the power log the square root of 3")[$10^("log" sqrt(3))$] ⓑ #math.equation(block: false, alt: "log 10 to the power −1")[$"log" 10^(−1)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the square root of 3")[$sqrt(3)$] ⓑ #math.equation(block: false, alt: "−1")[$−1$]
]

ⓐ #math.equation(block: false, alt: "e to the power ln 4")[$e^("ln" 4)$] ⓑ #math.equation(block: false, alt: "ln e squared")[$"ln" #h(0.2em) e^(2)$]

ⓐ #math.equation(block: false, alt: "e to the power ln 3")[$e^("ln" 3)$] ⓑ #math.equation(block: false, alt: "ln e to the power 7")[$"ln" #h(0.2em) e^(7)$]

#solutionbox[
ⓐ 3 ⓑ 7
]

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

#math.equation(block: true, alt: "log sub 4 6 x")[$"log"_(4) 6 x$]

#math.equation(block: true, alt: "log sub 5 8 y")[$"log"_(5) 8 y$]

#solutionbox[
#math.equation(block: true, alt: "log sub 5 8 plus log sub 5 y")[$"log"_(5) 8 + "log"_(5) y$]
]

#math.equation(block: true, alt: "log sub 2 32 x y")[$"log"_(2) 32 x y$]

#math.equation(block: true, alt: "log sub 3 81 x y")[$"log"_(3) 81 x y$]

#solutionbox[
#math.equation(block: true, alt: "4 plus log sub 3 x plus log sub 3 y")[$4 + "log"_(3) x + "log"_(3) y$]
]

#math.equation(block: true, alt: "log 100 x")[$"log" 100 x$]

#math.equation(block: true, alt: "log 1000 y")[$"log" 1000 y$]

#solutionbox[
#math.equation(block: true, alt: "3 plus log y")[$3 + "log" y$]
]

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

#math.equation(block: true, alt: "log sub 3 the fraction 3 over 8")[$"log"_(3) frac(3, 8)$]

#math.equation(block: true, alt: "log sub 6 the fraction 5 over 6")[$"log"_(6) frac(5, 6)$]

#solutionbox[
#math.equation(block: true, alt: "log sub 6 5 minus 1")[$"log"_(6) 5 − 1$]
]

#math.equation(block: true, alt: "log sub 4 the fraction 16 over y")[$"log"_(4) frac(16, y)$]

#math.equation(block: true, alt: "log sub 5 the fraction 125 over x")[$"log"_(5) frac(125, x)$]

#solutionbox[
#math.equation(block: true, alt: "3 minus log sub 5 x")[$3 − "log"_(5) x$]
]

#math.equation(block: true, alt: "log the fraction x over 10")[$"log" frac(x, 10)$]

#math.equation(block: true, alt: "log the fraction 10,000 over y")[$"log" frac(10","000, y)$]

#solutionbox[
#math.equation(block: true, alt: "4 minus log y")[$4 − "log" y$]
]

#math.equation(block: true, alt: "ln the fraction e cubed over 3")[$"ln" frac(e^(3), 3)$]

#math.equation(block: true, alt: "ln the fraction e to the power 4 over 16")[$"ln" frac(e^(4), 16)$]

#solutionbox[
#math.equation(block: true, alt: "4 minus ln 16")[$4 − "ln" 16$]
]

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

#math.equation(block: true, alt: "log sub 3 x squared")[$"log"_(3) x^(2)$]

#math.equation(block: true, alt: "log sub 2 x to the power 5")[$"log"_(2) x^(5)$]

#solutionbox[
#math.equation(block: true, alt: "5 log sub 2 x")[$5 "log"_(2) x$]
]

#math.equation(block: true, alt: "log x to the power −2")[$"log" x^(−2)$]

#math.equation(block: true, alt: "log x to the power −3")[$"log" x^(−3)$]

#solutionbox[
#math.equation(block: true, alt: "−3 log x")[$−3 "log" #h(0.2em) x$]
]

#math.equation(block: true, alt: "log sub 4 the square root of x")[$"log"_(4) sqrt(x)$]

#math.equation(block: true, alt: "log sub 5 the cube root of x")[$"log"_(5) root(3, x)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 3 log sub 5 x")[$frac(1, 3) "log"_(5) x$]
]

#math.equation(block: true, alt: "ln x to the power the square root of 3")[$"ln" #h(0.2em) x^(sqrt(3))$]

#math.equation(block: true, alt: "ln x to the power the cube root of 4")[$"ln" #h(0.2em) x^(root(3, 4))$]

#solutionbox[
#math.equation(block: true, alt: "the cube root of 4 ln x")[$root(3, 4) "ln" #h(0.2em) x$]
]

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

#math.equation(block: true, alt: "log sub 5 open parenthesis 4 x to the power 6 y to the power 4 close parenthesis")[$"log"_(5) ( 4 x^(6) y^(4) )$]

#math.equation(block: true, alt: "log sub 2 open parenthesis 3 x to the power 5 y cubed close parenthesis")[$"log"_(2) ( 3 x^(5) y^(3) )$]

#solutionbox[
#math.equation(block: true, alt: "log sub 2 3 plus 5 log sub 2 x plus 3 log sub 2 y")[$"log"_(2) 3 + 5 "log"_(2) x + 3 "log"_(2) y$]
]

#math.equation(block: true, alt: "log sub 3 open parenthesis the square root of 2 x squared close parenthesis")[$"log"_(3) ( sqrt(2) x^(2) )$]

#math.equation(block: true, alt: "log sub 5 open parenthesis the 4th root of 21 y cubed close parenthesis")[$"log"_(5) ( root(4, 21) y^(3) )$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 4 log sub 5 21 plus 3 log sub 5 y")[$frac(1, 4) "log"_(5) 21 + 3 "log"_(5) y$]
]

#math.equation(block: true, alt: "log sub 3 the fraction x y squared over z squared")[$"log"_(3) frac(x y^(2), z^(2))$]

#math.equation(block: true, alt: "log sub 5 the fraction 4 a b cubed c to the power 4 over d squared")[$"log"_(5) frac(4 a b^(3) c^(4), d^(2))$]

#solutionbox[
#math.equation(block: true, alt: "log sub 5 4 plus log sub 5 a plus 3 log sub 5 b")[$"log"_(5) 4 + "log"_(5) a + 3 "log"_(5) b$] #linebreak() #math.equation(block: true, alt: "plus 4 log sub 5 c minus 2 log sub 5 d")[$+ #h(0.2em) 4 "log"_(5) c − 2 "log"_(5) d$]
]

#math.equation(block: true, alt: "log sub 4 the fraction the square root of x over 16 y to the power 4")[$"log"_(4) frac(sqrt(x), 16 y^(4))$]

#math.equation(block: true, alt: "log sub 3 the fraction the cube root of x squared over 27 y to the power 4")[$"log"_(3) frac(root(3, x^(2)), 27 y^(4))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 3 log sub 3 x minus 3 minus 4 log sub 3 y")[$frac(2, 3) "log"_(3) x − 3 − 4 "log"_(3) y$]
]

#math.equation(block: true, alt: "log sub 2 the fraction the square root of 2 x plus y squared over z squared")[$"log"_(2) frac(sqrt(2 x + y^(2)), z^(2))$]

#math.equation(block: true, alt: "log sub 3 the fraction the square root of 3 x plus 2 y squared over 5 z squared")[$"log"_(3) frac(sqrt(3 x + 2 y^(2)), 5 z^(2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 2 log sub 3 open parenthesis 3 x plus 2 y squared close parenthesis minus log sub 3 5 minus 2 log sub 3 z")[$frac(1, 2) "log"_(3) ( 3 x + 2 y^(2) ) − "log"_(3) 5 − 2 "log"_(3) z$]
]

#math.equation(block: true, alt: "log sub 2 the 4th root of the fraction 5 x cubed over 2 y squared z to the power 4")[$"log"_(2) root(4, frac(5 x^(3), 2 y^(2) z^(4)))$]

#math.equation(block: true, alt: "log sub 5 the cube root of the fraction 3 x squared over 4 y cubed z")[$"log"_(5) root(3, frac(3 x^(2), 4 y^(3) z))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis log sub 5 3 plus 2 log sub 5 x minus log sub 5 4")[$frac(1, 3) \( "log"_(5) 3 + 2 "log"_(5) x − "log"_(5) 4$] #linebreak() #math.equation(block: true, alt: "minus 3 log sub 5 y minus log sub 5 z close parenthesis")[$− #h(0.2em) 3 "log"_(5) y − "log"_(5) z \)$]
]

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

#math.equation(block: true, alt: "log sub 6 4 plus log sub 6 9")[$"log"_(6) 4 + "log"_(6) 9$]

#math.equation(block: true, alt: "log 4 plus log 25")[$"log" 4 + "log" 25$]

#solutionbox[
2
]

#math.equation(block: true, alt: "log sub 2 80 minus log sub 2 5")[$"log"_(2) 80 − "log"_(2) 5$]

#math.equation(block: true, alt: "log sub 3 36 minus log sub 3 4")[$"log"_(3) 36 − "log"_(3) 4$]

#solutionbox[
2
]

#math.equation(block: true, alt: "log sub 3 4 plus log sub 3 open parenthesis x plus 1 close parenthesis")[$"log"_(3) 4 + "log"_(3) ( x + 1 )$]

#math.equation(block: true, alt: "log sub 2 5 minus log sub 2 open parenthesis x minus 1 close parenthesis")[$"log"_(2) 5 − "log"_(2) ( x − 1 )$]

#solutionbox[
#math.equation(block: true, alt: "log sub 2 the fraction 5 over x minus 1")[$"log"_(2) frac(5, x − 1)$]
]

#math.equation(block: true, alt: "log sub 7 3 plus log sub 7 x minus log sub 7 y")[$"log"_(7) 3 + "log"_(7) x − "log"_(7) y$]

#math.equation(block: true, alt: "log sub 5 2 minus log sub 5 x minus log sub 5 y")[$"log"_(5) 2 − "log"_(5) x − "log"_(5) y$]

#solutionbox[
#math.equation(block: true, alt: "log sub 5 the fraction 2 over x y")[$"log"_(5) frac(2, x y)$]
]

#math.equation(block: true, alt: "4 log sub 2 x plus 6 log sub 2 y")[$4 "log"_(2) x + 6 "log"_(2) y$]

#math.equation(block: true, alt: "6 log sub 3 x plus 9 log sub 3 y")[$6 "log"_(3) x + 9 "log"_(3) y$]

#solutionbox[
#math.equation(block: true, alt: "log sub 3 x to the power 6 y to the power 9")[$"log"_(3) x^(6) y^(9)$]
]

#math.equation(block: true, alt: "log sub 3 open parenthesis x squared minus 1 close parenthesis minus 2 log sub 3 open parenthesis x minus 1 close parenthesis")[$"log"_(3) ( x^(2) − 1 ) − 2 "log"_(3) ( x − 1 )$]

#math.equation(block: true, alt: "log open parenthesis x squared plus 2 x plus 1 close parenthesis minus 2 log open parenthesis x plus 1 close parenthesis")[$"log" ( x^(2) + 2 x + 1 ) − 2 "log" ( x + 1 )$]

#solutionbox[
0
]

#math.equation(block: true, alt: "4 log x minus 2 log y minus 3 log z")[$4 "log" #h(0.2em) x − 2 "log" y − 3 "log" z$]

#math.equation(block: true, alt: "3 ln x plus 4 ln y minus 2 ln z")[$3 "ln" #h(0.2em) x + 4 "ln" #h(0.2em) y − 2 "ln" #h(0.2em) z$]

#solutionbox[
#math.equation(block: true, alt: "ln the fraction x cubed y to the power 4 over z squared")[$"ln" frac(x^(3) y^(4), z^(2))$]
]

#math.equation(block: true, alt: "the fraction 1 over 3 log x minus 3 log open parenthesis x plus 1 close parenthesis")[$frac(1, 3) "log" #h(0.2em) x − 3 "log" ( x + 1 )$]

#math.equation(block: true, alt: "2 log open parenthesis 2 x plus 3 close parenthesis plus the fraction 1 over 2 log open parenthesis x plus 1 close parenthesis")[$2 "log" ( 2 x + 3 ) + frac(1, 2) "log" ( x + 1 )$]

#solutionbox[
#math.equation(block: true, alt: "log open parenthesis 2 x plus 3 close parenthesis squared times the square root of x plus 1")[$"log" attach(( 2 x + 3 ), t: 2) · sqrt(x + 1)$]
]

#strong[Use the Change-of-Base Formula]

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

#math.equation(block: true, alt: "log sub 3 42")[$"log"_(3) 42$]

#math.equation(block: true, alt: "log sub 5 46")[$"log"_(5) 46$]

#solutionbox[
#math.equation(block: true, alt: "2.379")[$2.379$]
]

#math.equation(block: true, alt: "log sub 12 87")[$"log"_(12) 87$]

#math.equation(block: true, alt: "log sub 15 93")[$"log"_(15) 93$]

#solutionbox[
#math.equation(block: true, alt: "1.674")[$1.674$]
]

#math.equation(block: true, alt: "log sub the square root of 2 17")[$"log"_(sqrt(2)) 17$]

#math.equation(block: true, alt: "log sub the square root of 3 21")[$"log"_(sqrt(3)) 21$]

#solutionbox[
#math.equation(block: true, alt: "5.542")[$5.542$]
]


==== Writing Exercises

Write the Product Property in your own words. Does it apply to each of the following? #math.equation(block: false, alt: "log sub a 5 x ,")[$"log"_(a) 5 x ,$]#math.equation(block: false, alt: "log sub a open parenthesis 5 plus x close parenthesis .")[$"log"_(a) ( 5 + x ) .$] Why or why not?

Write the Power Property in your own words. Does it apply to each of the following? #math.equation(block: false, alt: "log sub a x to the power p ,")[$"log"_(a) x^(p) ,$]#math.equation(block: false, alt: "open parenthesis log sub a x close parenthesis to the power r .")[$attach(( "log"_(a) x ), t: r) .$] Why or why not?

#solutionbox[
Answers will vary.
]

Use an example to show that #linebreak() #math.equation(block: false, alt: "log open parenthesis a plus b close parenthesis not equal to log a plus log b .")[$"log" ( a + b ) ≠ "log" a + "log" b .$]

Explain how to find the value of #math.equation(block: false, alt: "log sub 7 15")[$"log"_(7) 15$] using your calculator.

#solutionbox[
Answers will vary.
]


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has three rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads use the properties of logarithms and use the change of base formula. The rest of the cells are blank.], alt: "This table has three rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads use the properties of logarithms and use the change of base formula. The rest of the cells are blank.", caption: none)

ⓑ On a scale of #math.equation(block: false, alt: "1 minus 10 ,")[$1 − 10 ,$] how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?