6.3 Exponential Equations and Inequalities

In this section we will develop techniques for solving equations involving exponential functions. Suppose, for instance, we wanted to solve the equation $2^{x} = 128$ . After a moment's calculation, we find $128 = 2^{7}$ , so we have $2^{x} = 2^{7}$ . The one-to-one property of exponential functions, detailed in Theorem 6.4, tells us that $2^{x} = 2^{7}$ if and only if $x equals 7$ . This means that not only is $x equals 7$ a solution to $2^{x} = 2^{7}$ , it is the only solution. Now suppose we change the problem ever so slightly to $2^{x} = 129$ . We could use one of the inverse properties of exponentials and logarithms listed in Theorem 6.3 to write $129 = 2^{\log_{2} (129)}$ . We'd then have $2^{x} = 2^{\log_{2} (129)}$ , which means our solution is $x equals log sub 2 ⁡ open parenthesis 129 close parenthesis$ . This makes sense because, after all, the definition of $log sub 2 ⁡ open parenthesis 129 close parenthesis$ is `the exponent we put on $2$ to get $129$ .' Indeed we could have obtained this solution directly by rewriting the equation $2^{x} = 129$ in its logarithmic form $log sub 2 ⁡ open parenthesis 129 close parenthesis equals x$ . Either way, in order to get a reasonable decimal approximation to this number, we'd use the change of base formula, Theorem 6.7, to give us something more calculator friendly,¹ say $\log_{2} (129) = \frac{\ln (129)}{\ln (2)}$ . Another way to arrive at this answer is as follows

An exponential curve rising from left to right through the point (0, 1) is crossed by a horizontal line whose height c is set by a slider, and a small circle marks the single crossing point with a dashed segment dropping from it to the x-axis where the solution ln(c) divided by ln(a) is read. Raising the base a with its slider makes the curve climb more steeply so the crossing slides left, raising the line moves the crossing right, and dragging the line to zero or below makes the marked crossing disappear because the curve never reaches that height. Adjustable parameters: Base a (a) = 2, Line height c (c) = 5. Viewing window: x from -11.57 to 15.57, y from -2.39 to 14.39. — The equation a^x = c asks where the exponential curve y = a^x crosses the horizontal line y = c. The marked crossing is the solution, and its x-coordinate is exactly log_a(c) = ln(c)/ln(a), which the dashed drop to the x-axis reads off. Slide the base a and the line height c: a larger base reaches the line sooner, a higher line pushes the crossing to the right, and a line at or below zero never meets the curve at all, which is the graphical face of the domain of the logarithm.

$\begin{matrix} 2^{x} & = & 129 \\ \ln (2^{x}) & = & \ln (129) & Take the natural log of both sides. \\ x \ln (2) & = & \ln (129) & Power Rule \\ x & = & \frac{\ln (129)}{\ln (2)} \end{matrix}$

`Taking the natural log' of both sides is akin to squaring both sides: since $f (x) = \ln (x)$ is a function, as long as two quantities are equal, their natural logs are equal.² Also note that we treat $ln ⁡ open parenthesis 2 close parenthesis$ as any other non-zero real number and divide it through³ to isolate the variable $x$ . We summarize below the two common ways to solve exponential equations, motivated by our examples.

Steps for Solving an Equation involving Exponential Functions

Isolate the exponential function.
1. If convenient, express both sides with a common base and equate the exponents.
2. Otherwise, take the natural log of both sides of the equation and use the Power Rule.

Example 1

Solve the following equations. Check your answer graphically using a calculator.

$2^{3 x} = 16^{1 - x}$
$2000 = 1000 \cdot 3^{- 0.1 t}$
$9 \cdot 3^{x} = 7^{2 x}$
$75 = \frac{100}{1 + 3 e^{- 2 t}}$
$25^{x} = 5^{x} + 6$
$\frac{e^{x} - e^{- x}}{2} = 5$

Since $16$ is a power of $2$ , we can rewrite $2^{3 x} = 16^{1 - x}$ as $2^{3 x} = {(2^{4})}^{1 - x}$ . Using properties of exponents, we get $2^{3 x} = 2^{4 (1 - x)}$ . Using the one-to-one property of exponential functions, we get $3 x = 4 (1 - x)$ which gives $x = \frac{4}{7}$ . To check graphically, we set $f (x) = 2^{3 x}$ and $g (x) = 16^{1 - x}$ and see that they intersect at $x = \frac{4}{7} \approx 0.5714$ .
We begin solving $2000 = 1000 \cdot 3^{- 0.1 t}$ by dividing both sides by $1000$ to isolate the exponential which yields $3^{- 0.1 t} = 2$ . Since it is inconvenient to write $2$ as a power of $3$ , we use the natural log to get $\ln (3^{- 0.1 t}) = \ln (2)$ . Using the Power Rule, we get $- 0.1 t \ln (3) = \ln (2)$ , so we divide both sides by $minus 0.1 ⁢ ln ⁡ open parenthesis 3 close parenthesis$ to get $t = - \frac{\ln (2)}{0.1 \ln (3)} = - \frac{10 \ln (2)}{\ln (3)}$ . On the calculator, we graph $f ⁢ open parenthesis x close parenthesis equals 2000$ and $g (x) = 1000 \cdot 3^{- 0.1 x}$ and find that they intersect at $x = - \frac{10 \ln (2)}{\ln (3)} \approx - 6.3093$ .
Figure 6.42
Figure 6.43
$y = f (x) = 2^{3 x}$ and
$y = f (x) = 2000$ and
$y = g (x) = 16^{1 - x}$
$y = g (x) = 1000 \cdot 3^{- 0.1 x}$
We first note that we can rewrite the equation $9 \cdot 3^{x} = 7^{2 x}$ as $3^{2} \cdot 3^{x} = 7^{2 x}$ to obtain $3^{x + 2} = 7^{2 x}$ . Since it is not convenient to express both sides as a power of $3$ (or $7$ for that matter) we use the natural log: $\ln (3^{x + 2}) = \ln (7^{2 x})$ . The power rule gives $(x + 2) \ln (3) = 2 x \ln (7)$ . Even though this equation appears very complicated, keep in mind that $ln ⁡ open parenthesis 3 close parenthesis$ and $ln ⁡ open parenthesis 7 close parenthesis$ are just constants. The equation $(x + 2) \ln (3) = 2 x \ln (7)$ is actually a linear equation and as such we gather all of the terms with $x$ on one side, and the constants on the other. We then divide both sides by the coefficient of $x$ , which we obtain by factoring.
$\begin{matrix} (x + 2) \ln (3) & = & 2 x \ln (7) \\ x \ln (3) + 2 \ln (3) & = & 2 x \ln (7) \\ 2 \ln (3) & = & 2 x \ln (7) - x \ln (3) \\ 2 \ln (3) & = & x (2 \ln (7) - \ln (3)) & Factor. \\ x & = & \frac{2 \ln (3)}{2 \ln (7) - \ln (3)} \end{matrix}$
Graphing $f (x) = 9 \cdot 3^{x}$ and $g (x) = 7^{2 x}$ on the calculator, we see that these two graphs intersect at $x = \frac{2 \ln (3)}{2 \ln (7) - \ln (3)} \approx 0.7866$ .
Our objective in solving $75 = \frac{100}{1 + 3 e^{- 2 t}}$ is to first isolate the exponential. To that end, we clear denominators and get $75 (1 + 3 e^{- 2 t}) = 100$ . From this we get $75 + 225 e^{- 2 t} = 100$ , which leads to $225 e^{- 2 t} = 25$ , and finally, $e^{- 2 t} = \frac{1}{9}$ . Taking the natural log of both sides gives $\ln (e^{- 2 t}) = \ln (\frac{1}{9})$ . Since natural log is log base $e$ , $\ln (e^{- 2 t}) = - 2 t$ . We can also use the Power Rule to write $\ln (\frac{1}{9}) = - \ln (9)$ . Putting these two steps together, we simplify $\ln (e^{- 2 t}) = \ln (\frac{1}{9})$ to $- 2 t = - \ln (9)$ . We arrive at our solution, $t = \frac{\ln (9)}{2}$ which simplifies to $t equals ln ⁡ open parenthesis 3 close parenthesis$ . (Can you explain why?) The calculator confirms the graphs of $f ⁢ open parenthesis x close parenthesis equals 75$ and $g (x) = \frac{100}{1 + 3 e^{- 2 x}}$ intersect at $x = \ln (3) \approx 1.099$ .
Figure 6.44
Figure 6.45
$y = f (x) = 9 \cdot 3^{x}$ and
$y = f (x) = 75$ and
$y = g (x) = 7^{2 x}$
$y = g (x) = \frac{100}{1 + 3 e^{- 2 x}}$
We start solving $25^{x} = 5^{x} + 6$ by rewriting $25 equals 5 squared$ so that we have ${(5^{2})}^{x} = 5^{x} + 6$ , or $5^{2 x} = 5^{x} + 6$ . Even though we have a common base, having two terms on the right hand side of the equation foils our plan of equating exponents or taking logs. If we stare at this long enough, we notice that we have three terms with the exponent on one term exactly twice that of another. To our surprise and delight, we have a `quadratic in disguise'. Letting $u = 5^{x}$ , we have $u^{2} = {(5^{x})}^{2} = 5^{2 x}$ so the equation $5^{2 x} = 5^{x} + 6$ becomes $u^{2} = u + 6$ . Solving this as $u^{2} - u - 6 = 0$ gives $u = - 2$ or $u equals 3$ . Since $u = 5^{x}$ , we have $5^{x} = - 2$ or $5^{x} = 3$ . Since $5^{x} = - 2$ has no real solution, (Why not?) we focus on $5^{x} = 3$ . Since it isn't convenient to express $3$ as a power of $5$ , we take natural logs and get $\ln (5^{x}) = \ln (3)$ so that $x ⁢ ln ⁡ open parenthesis 5 close parenthesis equals ln ⁡ open parenthesis 3 close parenthesis$ or $x = \frac{\ln (3)}{\ln (5)}$ . On the calculator, we see the graphs of $f (x) = 25^{x}$ and $g (x) = 5^{x} + 6$ intersect at $x = \frac{\ln (3)}{\ln (5)} \approx 0.6826$ .
At first, it's unclear how to proceed with $\frac{e^{x} - e^{- x}}{2} = 5$ , besides clearing the denominator to obtain $e^{x} - e^{- x} = 10$ . Of course, if we rewrite $e^{- x} = \frac{1}{e^{x}}$ , we see we have another denominator lurking in the problem: $e^{x} - \frac{1}{e^{x}} = 10$ . Clearing this denominator gives us $e^{2 x} - 1 = 10 e^{x}$ , and once again, we have an equation with three terms where the exponent on one term is exactly twice that of another - a `quadratic in disguise.' If we let $u = e^{x}$ , then $u^{2} = e^{2 x}$ so the equation $e^{2 x} - 1 = 10 e^{x}$ can be viewed as $u^{2} - 1 = 10 u$ . Solving $u^{2} - 10 u - 1 = 0$ , we obtain by the quadratic formula $u = 5 \pm \sqrt{26}$ . From this, we have $e^{x} = 5 \pm \sqrt{26}$ . Since $5 - \sqrt{26} < 0$ , we get no real solution to $e^{x} = 5 - \sqrt{26}$ , but for $e^{x} = 5 + \sqrt{26}$ , we take natural logs to obtain $x = \ln (5 + \sqrt{26})$ . If we graph $f (x) = \frac{e^{x} - e^{- x}}{2}$ and $g ⁢ open parenthesis x close parenthesis equals 5$ , we see that the graphs intersect at $x = \ln (5 + \sqrt{26}) \approx 2.312$
Figure 6.46
Figure 6.47
$y = f (x) = 25^{x}$ and
$y = f (x) = \frac{e^{x} - e^{- x}}{2}$ and
$y = g (x) = 5^{x} + 6$
$y = g (x) = 5$

The authors would be remiss not to mention that Example 6.3.1 still holds great educational value. Much can be learned about logarithms and exponentials by verifying the solutions obtained in Example 6.3.1 analytically. For example, to verify our solution to $2000 = 1000 \cdot 3^{- 0.1 t}$ , we substitute $t = - \frac{10 \ln (2)}{\ln (3)}$ and obtain

$\begin{matrix} 2000 & \overset{?}{=} & 1000 \cdot 3^{- 0.1 (- \frac{10 \ln (2)}{\ln (3)})} \\ 2000 & \overset{?}{=} & 1000 \cdot 3^{\frac{\ln (2)}{\ln (3)}} \\ 2000 & \overset{?}{=} & 1000 \cdot 3^{\log_{3} (2)} & Change of Base \\ 2000 & \overset{?}{=} & 1000 \cdot 2 & Inverse Property \\ 2000 & \overset{✓}{=} & 2000 \end{matrix}$

The other solutions can be verified by using a combination of log and inverse properties. Some fall out quite quickly, while others are more involved. We leave them to the reader.

Since exponential functions are continuous on their domains, the Intermediate Value Theorem 3.1 applies. As with the algebraic functions in Section 5.3, this allows us to solve inequalities using sign diagrams as demonstrated below.

Example 2

Solve the following inequalities. Check your answer graphically using a calculator.

$2^{x^{2} - 3 x} - 16 \geq 0$
$\frac{e^{x}}{e^{x} - 4} \leq 3$
$x ⁢ e to the power 2 ⁢ x less than 4 ⁢ x$

Since we already have $0$ on one side of the inequality, we set $r (x) = 2^{x^{2} - 3 x} - 16$ . The domain of $r$ is all real numbers, so in order to construct our sign diagram, we need to find the zeros of $r$ . Setting $r ⁢ open parenthesis x close parenthesis equals 0$ gives $2^{x^{2} - 3 x} - 16 = 0$ or $2^{x^{2} - 3 x} = 16$ . Since $16 = 2^{4}$ we have $2^{x^{2} - 3 x} = 2^{4}$ , so by the one-to-one property of exponential functions, $x^{2} - 3 x = 4$ . Solving $x^{2} - 3 x - 4 = 0$ gives $x equals 4$ and $x = - 1$ . From the sign diagram, we see $r ⁢ open parenthesis x close parenthesis greater than or equal to 0$ on $(- \infty, - 1] \cup [4, \infty)$ , which corresponds to where the graph of $y = r (x) = 2^{x^{2} - 3 x} - 16$ , is on or above the $x$ -axis.
Figure 6.48
Figure 6.49
$y = r (x) = 2^{x^{2} - 3 x} - 16$
The first step we need to take to solve $\frac{e^{x}}{e^{x} - 4} \leq 3$ is to get $0$ on one side of the inequality. To that end, we subtract $3$ from both sides and get a common denominator
$\begin{matrix} \frac{e^{x}}{e^{x} - 4} & \leq & 3 \\ \frac{e^{x}}{e^{x} - 4} - 3 & \leq & 0 \\ \frac{e^{x}}{e^{x} - 4} - \frac{3 (e^{x} - 4)}{e^{x} - 4} & \leq & 0 & Common denomintors. \\ \frac{12 - 2 e^{x}}{e^{x} - 4} & \leq & 0 \end{matrix}$
We set $r (x) = \frac{12 - 2 e^{x}}{e^{x} - 4}$ and we note that $r$ is undefined when its denominator $e^{x} - 4 = 0$ , or when $e^{x} = 4$ . Solving this gives $x equals ln ⁡ open parenthesis 4 close parenthesis$ , so the domain of $r$ is $open parenthesis minus ∞, ln ⁡ open parenthesis 4 close parenthesis close parenthesis union open parenthesis ln ⁡ open parenthesis 4 close parenthesis, ∞ close parenthesis$ . To find the zeros of $r$ , we solve $r ⁢ open parenthesis x close parenthesis equals 0$ and obtain $12 - 2 e^{x} = 0$ . Solving for $e^{x}$ , we find $e^{x} = 6$ , or $x equals ln ⁡ open parenthesis 6 close parenthesis$ . When we build our sign diagram, finding test values may be a little tricky since we need to check values around $ln ⁡ open parenthesis 4 close parenthesis$ and $ln ⁡ open parenthesis 6 close parenthesis$ . Recall that the function $\ln (x)$ is increasing⁴ which means $ln ⁡ open parenthesis 3 close parenthesis less than ln ⁡ open parenthesis 4 close parenthesis less than ln ⁡ open parenthesis 5 close parenthesis less than ln ⁡ open parenthesis 6 close parenthesis less than ln ⁡ open parenthesis 7 close parenthesis$ . While the prospect of determining the sign of $r ⁢ open parenthesis ln ⁡ open parenthesis 3 close parenthesis close parenthesis$ may be very unsettling, remember that $e^{\ln (3)} = 3$ , so
$r (\ln (3)) = \frac{12 - 2 e^{\ln (3)}}{e^{\ln (3)} - 4} = \frac{12 - 2 (3)}{3 - 4} = - 6$
We determine the signs of $r ⁢ open parenthesis ln ⁡ open parenthesis 5 close parenthesis close parenthesis$ and $r ⁢ open parenthesis ln ⁡ open parenthesis 7 close parenthesis close parenthesis$ similarly.⁵ From the sign diagram, we find our answer to be $open parenthesis minus ∞, ln ⁡ open parenthesis 4 close parenthesis close parenthesis union open bracket ln ⁡ open parenthesis 6 close parenthesis, ∞ close parenthesis$ . Using the calculator, we see the graph of $f (x) = \frac{e^{x}}{e^{x} - 4}$ is below the graph of $g ⁢ open parenthesis x close parenthesis equals 3$ on $open parenthesis minus ∞, ln ⁡ open parenthesis 4 close parenthesis close parenthesis union open parenthesis ln ⁡ open parenthesis 6 close parenthesis, ∞ close parenthesis$ , and they intersect at $x = \ln (6) \approx 1.792$ .
Figure 6.50
Figure 6.51
$y = f (x) = \frac{e^{x}}{e^{x} - 4}$
$y = g (x) = 3$
As before, we start solving $x ⁢ e to the power 2 ⁢ x less than 4 ⁢ x$ by getting $0$ on one side of the inequality, $x e^{2 x} - 4 x < 0$ . We set $r (x) = x e^{2 x} - 4 x$ and since there are no denominators, even-indexed radicals, or logs, the domain of $r$ is all real numbers. Setting $r ⁢ open parenthesis x close parenthesis equals 0$ produces $x e^{2 x} - 4 x = 0$ . We factor to get $x (e^{2 x} - 4) = 0$ which gives $x equals 0$ or $e^{2 x} - 4 = 0$ . To solve the latter, we isolate the exponential and take logs to get $2 ⁢ x equals ln ⁡ open parenthesis 4 close parenthesis$ , or $x = \frac{\ln (4)}{2} = \ln (2)$ . (Can you explain the last equality using properties of logs?) As in the previous example, we need to be careful about choosing test values. Since $ln ⁡ open parenthesis 1 close parenthesis equals 0$ , we choose $\ln (\frac{1}{2})$ , $\ln (\frac{3}{2})$ and $ln ⁡ open parenthesis 3 close parenthesis$ . Evaluating,⁶ we get
$\begin{matrix} r (\ln (\frac{1}{2})) & = & \ln (\frac{1}{2}) e^{2 \ln (\frac{1}{2})} - 4 \ln (\frac{1}{2}) \\ = & \ln (\frac{1}{2}) e^{\ln {(\frac{1}{2})}^{2}} - 4 \ln (\frac{1}{2}) & Power Rule \\ = & \ln (\frac{1}{2}) e^{\ln (\frac{1}{4})} - 4 \ln (\frac{1}{2}) \\ = & \frac{1}{4} \ln (\frac{1}{2}) - 4 \ln (\frac{1}{2}) = - \frac{15}{4} \ln (\frac{1}{2}) \end{matrix}$
Since $\frac{1}{2} < 1$ , $\ln (\frac{1}{2}) < 0$ and we get $r (\ln (\frac{1}{2}))$ is $(+)$ , so $r ⁢ open parenthesis x close parenthesis less than 0$ on $open parenthesis 0, ln ⁡ open parenthesis 2 close parenthesis close parenthesis$ . The calculator confirms that the graph of $f (x) = x e^{2 x}$ is below the graph of $g ⁢ open parenthesis x close parenthesis equals 4 ⁢ x$ on these intervals.⁷
Figure 6.52
Figure 6.53
$y = f (x) = x e^{2 x}$ and $y = g (x) = 4 x$

Example 3

Recall from Example 6.1.2 that the temperature of coffee $T$ (in degrees Fahrenheit) $t$ minutes after it is served can be modeled by $T (t) = 70 + 90 e^{- 0.1 t}$ . When will the coffee be warmer than $100 degrees ⁢ F$ ?

We need to find when $T ⁢ open parenthesis t close parenthesis greater than 100$ , or in other words, we need to solve the inequality $70 + 90 e^{- 0.1 t} > 100$ . Getting $0$ on one side of the inequality, we have $90 e^{- 0.1 t} - 30 > 0$ , and we set $r (t) = 90 e^{- 0.1 t} - 30$ . The domain of $r$ is artificially restricted due to the context of the problem to $open bracket 0, ∞ close parenthesis$ , so we proceed to find the zeros of $r$ . Solving $90 e^{- 0.1 t} - 30 = 0$ results in $e^{- 0.1 t} = \frac{1}{3}$ so that $t = - 10 \ln (\frac{1}{3})$ which, after a quick application of the Power Rule leaves us with $t equals 10 ⁢ ln ⁡ open parenthesis 3 close parenthesis$ . If we wish to avoid using the calculator to choose test values, we note that since $1 less than 3$ , $0 equals ln ⁡ open parenthesis 1 close parenthesis less than ln ⁡ open parenthesis 3 close parenthesis$ so that $10 ⁢ ln ⁡ open parenthesis 3 close parenthesis greater than 0$ . So we choose $t equals 0$ as a test value in $open bracket 0, 10 ⁢ ln ⁡ open parenthesis 3 close parenthesis close parenthesis$ . Since $3 less than 4$ , $10 ⁢ ln ⁡ open parenthesis 3 close parenthesis less than 10 ⁢ ln ⁡ open parenthesis 4 close parenthesis$ , so the latter is our choice of a test value for the interval $open parenthesis 10 ⁢ ln ⁡ open parenthesis 3 close parenthesis, ∞ close parenthesis$ . Our sign diagram is below, and next to it is our graph of $y = T (t)$ from Example 6.1.2 with the horizontal line $y equals 100$ .

Figure 6.54

Figure 6.55

y = T (t)

In order to interpret what this means in the context of the real world, we need a reasonable approximation of the number $10 ⁢ ln ⁡ open parenthesis 3 close parenthesis approximately equals 10.986$ . This means it takes approximately $11$ minutes for the coffee to cool to $100 degrees ⁢ F$ . Until then, the coffee is warmer than that.⁸

We close this section by finding the inverse of a function which is a composition of a rational function with an exponential function.

Example 4

The function $f (x) = \frac{5 e^{x}}{e^{x} + 1}$ is one-to-one. Find a formula for $f^{- 1} (x)$ and check your answer graphically using your calculator.

We start by writing $y = f (x)$ , and interchange the roles of $x$ and $y$ . To solve for $y$ , we first clear denominators and then isolate the exponential function.

$\begin{matrix} y & = & \frac{5 e^{x}}{e^{x} + 1} \\ x & = & \frac{5 e^{y}}{e^{y} + 1} & Switch x and y \\ x (e^{y} + 1) & = & 5 e^{y} \\ x e^{y} + x & = & 5 e^{y} \\ x & = & 5 e^{y} - x e^{y} \\ x & = & e^{y} (5 - x) \\ e^{y} & = & \frac{x}{5 - x} \\ \ln (e^{y}) & = & \ln (\frac{x}{5 - x}) \\ y & = & \ln (\frac{x}{5 - x}) \end{matrix}$

We claim $f^{- 1} (x) = \ln (\frac{x}{5 - x})$ . To verify this analytically, we would need to verify the compositions $(f^{- 1} \circ f) (x) = x$ for all $x$ in the domain of $f$ and that $(f \circ f^{- 1}) (x) = x$ for all $x$ in the domain of $f^{- 1}$ . We leave this to the reader. To verify our solution graphically, we graph $y = f (x) = \frac{5 e^{x}}{e^{x} + 1}$ and $y = g (x) = \ln (\frac{x}{5 - x})$ on the same set of axes and observe the symmetry about the line $y = x$ . Note the domain of $f$ is the range of $g$ and vice-versa.

Figure 6.56

$y = f (x) = \frac{5 e^{x}}{e^{x} + 1}$ and $y = g (x) = \ln (\frac{x}{5 - x})$

Exercises

In Exercises -, solve the equation analytically.

$2^{4 x} = 8$
$3^{(x - 1)} = 27$
$5^{2 x - 1} = 125$
$4^{2 x} = \frac{1}{2}$
$8^{x} = \frac{1}{128}$
$2^{(x^{3} - x)} = 1$
$3^{7 x} = 81^{4 - 2 x}$
$9 \cdot 3^{7 x} = {(\frac{1}{9})}^{2 x}$
$3^{2 x} = 5$
$5^{- x} = 2$
$5^{x} = - 2$
$3^{(x - 1)} = 29$
${(1.005)}^{12 x} = 3$
$e^{- 5730 k} = \frac{1}{2}$
$2000 e^{0.1 t} = 4000$
$500 (1 - e^{2 x}) = 250$
$70 + 90 e^{- 0.1 t} = 75$
$30 - 6 e^{- 0.1 x} = 20$
$\frac{100 e^{x}}{e^{x} + 2} = 50$
$\frac{5000}{1 + 2 e^{- 3 t}} = 2500$
$\frac{150}{1 + 29 e^{- 0.8 t}} = 75$
$25 {(\frac{4}{5})}^{x} = 10$
$e^{2 x} = 2 e^{x}$
$7 e^{2 x} = 28 e^{- 6 x}$
$3^{(x - 1)} = 2^{x}$
$3^{(x - 1)} = {(\frac{1}{2})}^{(x + 5)}$
$7^{3 + 7 x} = 3^{4 - 2 x}$
$e^{2 x} - 3 e^{x} - 10 = 0$
$e^{2 x} = e^{x} + 6$
$4^{x} + 2^{x} = 12$
$e^{x} - 3 e^{- x} = 2$
$e^{x} + 15 e^{- x} = 8$
$3^{x} + 25 \cdot 3^{- x} = 10$

In Exercises -, solve the inequality analytically.

$e to the power x greater than 53$
$1000 ⁢ open parenthesis 1.005 close parenthesis to the power 12 ⁢ t greater than or equal to 3000$
$2^{(x^{3} - x)} < 1$
$25 {(\frac{4}{5})}^{x} \geq 10$
$\frac{150}{1 + 29 e^{- 0.8 t}} \leq 130$
$70 + 90 e^{- 0.1 t} \leq 75$

In Exercises -, use your calculator to help you solve the equation or inequality.

$2^{x} = x^{2}$
$e^{x} = \ln (x) + 5$
$e^{\sqrt{x}} = x + 1$
$e^{- x} - x e^{- x} \geq 0$
$3^{(x - 1)} < 2^{x}$
$e^{x} < x^{3} - x$
Since $f (x) = \ln (x)$ is a strictly increasing function, if $0 less than a less than b$ then $\ln (a) < \ln (b)$ . Use this fact to solve the inequality $e^{(3 x - 1)} > 6$ without a sign diagram. Use this technique to solve the inequalities in Exercises -. (NOTE: Isolate the exponential function first!)
Compute the inverse of $f (x) = \frac{e^{x} - e^{- x}}{2}$ . State the domain and range of both $f$ and $f^{- 1}$ .
In Example 6.3.4, we found that the inverse of $f (x) = \frac{5 e^{x}}{e^{x} + 1}$ was $f^{- 1} (x) = \ln (\frac{x}{5 - x})$ but we left a few loose ends for you to tie up.
1. Show that $(f^{- 1} \circ f) (x) = x$ for all $x$ in the domain of $f$ and that $(f \circ f^{- 1}) (x) = x$ for all $x$ in the domain of $f^{- 1}$ .
2. Find the range of $f$ by finding the domain of $f^{- 1}$ .
3. Let $g (x) = \frac{5 x}{x + 1}$ and $h (x) = e^{x}$ . Show that $f equals g ∘ h$ and that ${(g \circ h)}^{- 1} = h^{- 1} \circ g^{- 1}$ . (We know this is true in general by Exercise in Section 5.2, but it's nice to see a specific example of the property.)
With the help of your classmates, solve the inequality $e^{x} > x^{n}$ for a variety of natural numbers $n$ . What might you conjecture about the “speed” at which $f (x) = e^{x}$ grows versus any polynomial?

Answers

$x = \frac{3}{4}$
$x equals 4$
$x equals 2$
$x = - \frac{1}{4}$
$x = - \frac{7}{3}$
$x = - 1, 0, 1$
$x = \frac{16}{15}$
$x = - \frac{2}{11}$
$x = \frac{\ln (5)}{2 \ln (3)}$
$x = - \frac{\ln (2)}{\ln (5)}$
No solution.
$x = \frac{\ln (29) + \ln (3)}{\ln (3)}$
$x = \frac{\ln (3)}{12 \ln (1.005)}$
$k = \frac{\ln (\frac{1}{2})}{- 5730} = \frac{\ln (2)}{5730}$
$t = \frac{\ln (2)}{0.1} = 10 \ln (2)$
$x = \frac{1}{2} \ln (\frac{1}{2}) = - \frac{1}{2} \ln (2)$
$t = \frac{\ln (\frac{1}{18})}{- 0.1} = 10 \ln (18)$
$x = - 10 \ln (\frac{5}{3}) = 10 \ln (\frac{3}{5})$
$x equals ln ⁡ open parenthesis 2 close parenthesis$
$t = \frac{1}{3} \ln (2)$
$t = \frac{\ln (\frac{1}{29})}{- 0.8} = \frac{5}{4} \ln (29)$
$x = \frac{\ln (\frac{2}{5})}{\ln (\frac{4}{5})} = \frac{\ln (2) - \ln (5)}{\ln (4) - \ln (5)}$
$x equals ln ⁡ open parenthesis 2 close parenthesis$
$x = - \frac{1}{8} \ln (\frac{1}{4}) = \frac{1}{4} \ln (2)$
$x = \frac{\ln (3)}{\ln (3) - \ln (2)}$
$x = \frac{\ln (3) + 5 \ln (\frac{1}{2})}{\ln (3) - \ln (\frac{1}{2})} = \frac{\ln (3) - 5 \ln (2)}{\ln (3) + \ln (2)}$
$x = \frac{4 \ln (3) - 3 \ln (7)}{7 \ln (7) + 2 \ln (3)}$
$x equals ln ⁡ open parenthesis 5 close parenthesis$
$x equals ln ⁡ open parenthesis 3 close parenthesis$
$x = \frac{\ln (3)}{\ln (2)}$
$x equals ln ⁡ open parenthesis 3 close parenthesis$
$x equals ln ⁡ open parenthesis 3 close parenthesis$ , $ln ⁡ open parenthesis 5 close parenthesis$
$x = \frac{\ln (5)}{\ln (3)}$
$open parenthesis ln ⁡ open parenthesis 53 close parenthesis, ∞ close parenthesis$
$[\frac{\ln (3)}{12 \ln (1.005)}, \infty)$
$(- \infty, - 1) \cup (0, 1)$
$(- \infty, \frac{\ln (\frac{2}{5})}{\ln (\frac{4}{5})}] = (- \infty, \frac{\ln (2) - \ln (5)}{\ln (4) - \ln (5)}]$
$(- \infty, \frac{\ln (\frac{2}{377})}{- 0.8}] = (- \infty, \frac{5}{4} \ln (\frac{377}{2})]$
$[\frac{\ln (\frac{1}{18})}{- 0.1}, \infty) = [10 \ln (18), \infty)$
$x \approx - 0.76666, x = 2, x = 4$
$x \approx 0.01866, x \approx 1.7115$
$x equals 0$
$open parenthesis minus ∞, 1 close bracket$
$\approx (- \infty, 2.7095)$
$approximately equals open parenthesis 2.3217, 4.3717 close parenthesis$
$x > \frac{1}{3} (\ln (6) + 1)$
$f^{- 1} = \ln (x + \sqrt{x^{2} + 1})$ . Both $f$ and $f^{- 1}$ have domain $open parenthesis minus ∞, ∞ close parenthesis$ and range $open parenthesis minus ∞, ∞ close parenthesis$ .

You can use natural logs or common logs. We choose natural logs. (In Calculus, you'll learn these are the most `mathy' of the logarithms.) ↩
This is also the `if' part of the statement $\log_{b} (u) = \log_{b} (w)$ if and only if $u = w$ in Theorem 6.4. ↩
Please resist the temptation to divide both sides by ` $\ln$ ' instead of $ln ⁡ open parenthesis 2 close parenthesis$ . Just like it wouldn't make sense to divide both sides by the square root symbol ` $\sqrt{}$ ' when solving $x \sqrt{2} = 5$ , it makes no sense to divide by ` $\ln$ '. ↩
This is because the base of $\ln (x)$ is $e greater than 1$ . If the base $b$ were in the interval $0 less than b less than 1$ , then $\log_{b} (x)$ would decreasing. ↩
We could, of course, use the calculator, but what fun would that be? ↩
A calculator can be used at this point. As usual, we proceed without apologies, with the analytical method. ↩
Note: $ln ⁡ open parenthesis 2 close parenthesis approximately equals 0.693$ . ↩
Critics may point out that since we needed to use the calculator to interpret our answer anyway, why not use it earlier to simplify the computations? It is a fair question which we answer unfairly: it's our book. ↩

Adapted from Precalculus, 3rd corrected edition, by Carl Stitz and Jeff Zeager (stitz-zeager.com), licensed under CC BY-NC-SA 3.0. Changes were made: reformatted as an accessible XYZ web edition. License: CC-BY-NC-SA-3.0.