10.5 Solve Exponential and Logarithmic Equations

$${\text{log}}_{4}64=2{\text{log}}_{4}x$$

$$\text{log}49=2\text{log}x$$

$$3{\text{log}}_{3}x={\text{log}}_{3}27$$

$$3{\text{log}}_{6}x={\text{log}}_{6}64$$

$${\text{log}}_5\left(4x-2\right)={\text{log}}_{5}10$$

$${\text{log}}_3\left(x^2+3\right)={\text{log}}_{3}4x$$

$${\text{log}}_{3}x+{\text{log}}_{3}x=2$$

$${\text{log}}_{4}x+{\text{log}}_{4}x=3$$

$${\text{log}}_{2}x+{\text{log}}_2\left(x-3\right)=2$$

$${\text{log}}_{3}x+{\text{log}}_3\left(x+6\right)=3$$

$$\text{log}x+\text{log}\left(x+3\right)=1$$

$$\text{log}x+\text{log}\left(x-15\right)=2$$

$$\text{log}\left(x+4\right)-\text{log}\left(5x+12\right)=\text{−}\text{log}x$$

$$\text{log}\left(x-1\right)-\text{log}\left(x+3\right)=\text{log}\frac{1}{x}$$

$${\text{log}}_5\left(x+3\right)+{\text{log}}_5\left(x-6\right)={\text{log}}_{5}10$$

$${\text{log}}_5\left(x+1\right)+{\text{log}}_5\left(x-5\right)={\text{log}}_{5}7$$

$${\text{log}}_3\left(2x-1\right)={\text{log}}_3\left(x+3\right)+{\text{log}}_{3}3$$

$$\text{log}\left(5x+1\right)=\text{log}\left(x+3\right)+\text{log}2$$

$$3^x=89$$

$$2^x=74$$

$$5^x=110$$

$$4^x=112$$

$$e^x=16$$

$$e^x=8$$

$${\left(\frac{1}{2}\right)}^x=6$$

$${\left(\frac{1}{3}\right)}^x=8$$

$$4e^{x+1}=16$$

$$3e^{x+2}=9$$

$$6e^{2x}=24$$

$$2e^{3x}=32$$

$$\frac{1}{4}{e}^x=3$$

$$\frac{1}{3}{e}^x=2$$

$$e^{x+1}+2=16$$

$$e^{x-1}+4=12$$

$$3^{3x+1}=81$$

$$6^{4x-17}=216$$

$$\frac{e^{x^2}}{e^{14}}=e^{5x}$$

$$\frac{e^{x^2}}{e^x}=e^{20}$$

$${\text{log}}_{a}64=2$$

$${\text{log}}_{a}81=4$$

$$\text{ln}x=\mathrm{-8}$$

$$\text{ln}x=9$$

$${\text{log}}_5(3x-8)=2$$

$${\text{log}}_4(7x+15)=3$$

$$\text{ln}{e}^{5x}=30$$

$$\text{ln}{e}^{6x}=18$$

$$3\text{log}x=\text{log}125$$

$$7{\text{log}}_{3}x={\text{log}}_{3}128$$

$${\text{log}}_{6}x+{\text{log}}_6(x-5)={\text{log}}_{6}24$$

$${\text{log}}_{9}x+{\text{log}}_9(x-4)={\text{log}}_{9}12$$

$${\text{log}}_2\left(x+2\right)-{\text{log}}_2(2x+9)=\text{−}{\text{log}}_{2}x$$

$${\text{log}}_6\left(x+1\right)-{\text{log}}_6(4x+10)={\text{log}}_6\frac{1}{x}$$

$$6^x=91$$

$${\left(\frac{1}{2}\right)}^x=10$$

$$7e^{x-3}=35$$

$$8e^{x+5}=56$$

Sung Lee invests $\text{\$}\mathrm{5,000}$ at age 18. He hopes the investments will be worth $\text{\$}\mathrm{10,000}$ when he turns 25. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?

Alice invests $\text{\$}\mathrm{15,000}$ at age 30 from the signing bonus of her new job. She hopes the investments will be worth $\text{\$}\mathrm{30,000}$ when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

Coralee invests $\text{\$}\mathrm{5,000}$ in an account that compounds interest monthly and earns $7\text{\%}.$ How long will it take for her money to double?

Simone invests $\text{\$}\mathrm{8,000}$ in an account that compounds interest quarterly and earns $5\text{\%}.$ How long will it take for his money to double?

Researchers recorded that a certain bacteria population declined from 100,000 to 100 in 24 hours. At this rate of decay, how many bacteria will there be in 16 hours?

Researchers recorded that a certain bacteria population declined from 800,000 to 500,000 in 6 hours after the administration of medication. At this rate of decay, how many bacteria will there be in 24 hours?

A virus takes 6 days to double its original population $\left(A=2A_0\right).$ How long will it take to triple its population?

A bacteria doubles its original population in 24 hours $\left(A=2A_0\right).$ How big will its population be in 72 hours?

Carbon-14 is used for archeological carbon dating. Its half-life is 5,730 years. How much of a 100-gram sample of Carbon-14 will be left in 1000 years?

Radioactive technetium-99m is often used in diagnostic medicine as it has a relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is 6 hours, how much of the radioactive material form a 0.5 ml injection will be in the body in 24 hours?

Explain the method you would use to solve these equations: $3^{x+1}=81,$ $3^{x+1}=75.$ Does your method require logarithms for both equations? Why or why not?

What is the difference between the equation for exponential growth versus the equation for exponential decay?

$f\left(x\right)=7x-2$ and
$g\left(x\right)=5x+1$

$f\left(x\right)=4x$ and
$g\left(x\right)=x^2+3x$

For functions
$f\left(x\right)=3x^2+2$ and
$g\left(x\right)=4x-3,$ find
ⓐ $\left(f\circ g\right)\left(\mathrm{-3}\right)$
ⓑ $\left(g\circ f\right)\left(\mathrm{-2}\right)$
ⓒ $\left(f\circ f\right)\left(\mathrm{-1}\right)$

For functions
$f\left(x\right)=2x^3+5$ and
$g\left(x\right)=3x^2-7,$ find
ⓐ $\left(f\circ g\right)\left(\mathrm{-1}\right)$
ⓑ $\left(g\circ f\right)\left(\mathrm{-2}\right)$
ⓒ $\left(g\circ g\right)\left(1\right)$

$$\{\left(\mathrm{-3},\mathrm{-5}\right),\left(\mathrm{-2},\mathrm{-4}\right),\left(\mathrm{-1},\mathrm{-3}\right),\left(0,\mathrm{-2}\right),$$
$$\left(\mathrm{-1},\mathrm{-1}\right),\left(\mathrm{-2},0\right),\left(\mathrm{-3},1\right)\}$$

$$\{\left(\mathrm{-3},0\right),\left(\mathrm{-2},\mathrm{-2}\right),\left(\mathrm{-1},0\right),\left(0,1\right),$$
$$\left(1,2\right),\left(2,1\right),\left(3,\mathrm{-1}\right)\}$$

$$\{\left(\mathrm{-3},3\right),\left(\mathrm{-2},1\right),\left(\mathrm{-1},\mathrm{-1}\right),\left(0,\mathrm{-3}\right),$$
$$\left(1,\mathrm{-5}\right),\left(2,\mathrm{-4}\right),\left(3,\mathrm{-2}\right)\}$$

ⓐ

ⓐ

$$\left\{\left(\mathrm{-3},10\right),\left(\mathrm{-2},5\right),\left(\mathrm{-1},2\right),\left(0,1\right)\right\}$$

$f\left(x\right)=3x+7$ and
$g\left(x\right)=\frac{x-7}{3}$

$f\left(x\right)=2x+9$ and
$g\left(x\right)=\frac{x+9}{2}$

$$f\left(x\right)=6x-11$$

$$f\left(x\right)=x^3+13$$

$$f\left(x\right)=\frac{1}{x+5}$$

$$f\left(x\right)=\sqrt[5]{x-1}$$

$$f\left(x\right)=4^x$$

$$f\left(x\right)={\left(\frac{1}{5}\right)}^x$$

$$g\left(x\right)={\left(0.75\right)}^x$$

$$g\left(x\right)=3^{x+2}$$

$$f\left(x\right)={(2.3)}^x-3$$

$$f\left(x\right)=e^x+5$$

$$f\left(x\right)=\text{−}{e}^x$$

$$3^{5x-6}=81$$

$$2^{x^2}=16$$

$$9^x=27$$

$$5^{x^2+2x}=\frac{1}{5}$$

$$e^{4x}·e^7=e^{19}$$

$$\frac{e^{x^2}}{e^{15}}=e^{2x}$$

Felix invested $\text{\$}\mathrm{12,000}$ in a savings account. If the interest rate is $4\text{\%}$ how much will be in the account in 12 years by each method of compounding?

ⓐ compound quarterly
ⓑ compound monthly
ⓒ compound continuously.

Sayed deposits $\text{\$}\mathrm{20,000}$ in an investment account. What will be the value of his investment in 30 years if the investment is earning $7\text{\%}$ per year and is compounded continuously?

A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. She starts her experiment with 150 of the bacteria that grows at a rate of $15\text{\%}$ per hour. She will check on the bacteria every 24 hours. How many bacteria will he find in 24 hours?

In the last five years the population of the United States has grown at a rate of $0.7\text{\%}$ per year to about 318,900,000. If this rate continues, what will be the population in 5 more years?

$$5^4=625$$

$${10}^{\mathrm{-3}}=\frac{1}{\mathrm{1,000}}$$

$${63}^{\frac{1}{5}}=\sqrt[5]{63}$$

$$e^y=16$$

$$7={\text{log}}_{2}128$$

$$5=\text{log}\mathrm{100,000}$$

$$4=\text{ln}x$$

$${\text{log}}_{x}125=3$$

$${\text{log}}_{7}x=\mathrm{-2}$$

$${\text{log}}_{\frac{1}{2}}\frac{1}{16}=x$$

$${\text{log}}_{2}32$$

$${\text{log}}_{8}1$$

$${\text{log}}_3\frac{1}{9}$$

$$y={\text{log}}_{5}x$$

$$y={\text{log}}_{\frac{1}{4}}x$$

$$y={\text{log}}_{0.8}x$$

$${\text{log}}_{a}36=5$$

$$\text{ln}x=\mathrm{-3}$$

$${\text{log}}_2\left(5x-7\right)=3$$

$$\text{ln}{e}^{3x}=24$$

$$\text{log}(x^2-21)=2$$

What is the decibel level of a train whistle with intensity ${10}^{\mathrm{-3}}$ watts per square inch?

ⓐ ${\text{log}}_{7}1$ ⓑ ${\text{log}}_{12}12$

ⓐ $5^{{\text{log}}_{5}13}$ ⓑ ${\text{log}}_3{3}^{\mathrm{-9}}$

ⓐ ${10}^{\text{log}\sqrt{5}}$ ⓑ $\text{log}{10}^{\mathrm{-3}}$

ⓐ $e^{\text{ln}8}$ ⓑ $\text{ln}{e}^5$

$${\text{log}}_4\left(64xy\right)$$

$$\text{log}\mathrm{10,000}m$$

$${\text{log}}_7\frac{49}{y}$$

$$\text{ln}\frac{e^5}{2}$$

$$\text{log}{x}^{\mathrm{-9}}$$

$${\text{log}}_4\sqrt[7]{z}$$

$${\text{log}}_3\left(\sqrt{4}{x}^7{y}^8\right)$$

$${\log }_5\frac{8a^2{b}^{6}c}{d^3}$$

$$\text{ln}\frac{\sqrt{3x^2-y^2}}{z^4}$$

$${\text{log}}_6\sqrt[3]{\frac{7x^2}{6y^3{z}^5}}$$

$${\text{log}}_{2}56-{\text{log}}_{2}7$$

$$3{\text{log}}_{3}x+7{\text{log}}_{3}y$$

$${\text{log}}_5\left(x^2-16\right)-2{\text{log}}_5\left(x+4\right)$$

$$\frac{1}{4}\text{log}y-2\text{log}\left(y-3\right)$$

$${\text{log}}_{5}97$$

$${\text{log}}_{\sqrt{3}}16$$

$$3{\text{log}}_{5}x={\text{log}}_{5}216$$

$${\text{log}}_{2}x+{\text{log}}_2\left(x-2\right)=3$$

$$\text{log}\left(x-1\right)-\text{log}\left(3x+5\right)=\text{−}\text{log}x$$

$${\text{log}}_4\left(x-2\right)+{\text{log}}_4(x+5)={\text{log}}_{4}8$$

$$\text{ln}\left(3x-2\right)=\text{ln}\left(x+4\right)+\text{ln}2$$

$$2^x=101$$

$$e^x=23$$

$${\left(\frac{1}{3}\right)}^x=7$$

$$7e^{x+3}=28$$

$$e^{x-4}+8=23$$

Jerome invests $\text{\$}\mathrm{18,000}$ at age 17. He hopes the investments will be worth $\text{\$}\mathrm{30,000}$ when he turns 26. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?

Elise invests $\text{\$}4500$ in an account that compounds interest monthly and earns $6\text{\%}.$ How long will it take for her money to double?

Researchers recorded that a certain bacteria population grew from 100 to 300 in 8 hours. At this rate of growth, how many bacteria will there be in 24 hours?

Mouse populations can double in 8 months $\left(A=2A_0\right).$ How long will it take for a mouse population to triple?

The half-life of radioactive iodine is 60 days. How much of a 50 mg sample will be left in 40 days?