Exponential and Logarithmic Equations

Learning Objectives

Solve Exponential Equations. (IA 10.2.2)
Solve Logarithmic Equations. (IA 10.3.4)

Objective 1: Solve Exponential Equations. (IA 10.2.2)

Equations that include an exponential expression $a^{x}$ are called exponential equations. There are two types of exponential equations: those with the common base on each side, and those without a common base.

Type 1: Possible common base on each side: Use properties of exponents to rewrite each side with a common base. Use base-exponent property to set exponents equal to each other and solve for x.

Type 2: No possible common base: Use properties of exponents to rewrite each side in terms of one exponential expression. Take the log or ln of each side and use the power rule to bring down the power. Solve the remaining equation for x.

Example 1

Solving Exponential Equations.

Solve: $3^{2 x - 5} = 27 .$

Solution

.
Is here a common base?	Yes, both 3 and 27 can be rewritten as powers of 3.
Write both sides of the equation with the same base.	$3^{2 x - 5} = 3^{3}$
Since the bases are the same, the exponents must be equal.
Write a new equation by setting the exponents equal.	$2 x - 5 = 3$
Solve the equation.	$2 x = 8$
	$x = 4$
Check the solution by substituting x=4 into the original equation.	$3^{2 (4) - 5} = 27$
	$27 = 27,$ True

Example 2

Solve $3 e^{x + 2} = 24$ . Find the exact answer and then approximate it to three decimal places.

Solution

.
Rewriting with a common base is not possible.
Isolate the exponential by dividing both sides by 3.	$e^{x + 2} = 8$
Take the natural logarithm of both sides.	$\ln e^{x + 2} = \ln 8$
Use the Power Property to get the x as a factor, not an exponent.	$(x + 2) \ln e = \ln 8$
Use the property $\ln e = 1$ to simplify.	$x + 2 = \ln 8$
Solve the equation. Find the exact answer.	$x = \ln 8 -2$
Approximate the answer.	$x = 0.079$

Practice Makes Perfect

Solve. Find the exact answer and then approximate it to three decimal places.

$4^{2 x - 3} = \frac{1}{16}$

$5 (3^{x}) = 20$

Objective 2: Solving Logarithmic Equations. (IA 10.3.4)

There are two types of logarithmic equations: those with log terms on just one side of the equation or those with log terms on each side of the equation. Since the domain of logarithmic functions is positive numbers only, make sure to check the solutions.

Type 1: Log terms on one side of the equation: Use properties of logs to rewrite a side with just one log term. Convert to exponential notation and solve for x.

If $\log_{a} x = y$ then $x = a^{y}$ .

Type 2: Log terms on both sides of equation: First, use log properties to rewrite each side in terms of a single log expression, if necessary. Use the one-to-one property of logarithmic equality to set arguments equal to one another. Solve the resulting equation for x.

Example 3

Solving logarithmic equations.

Solve: $\log_{2} (3 x - 5) = 4$

Solution

.
Rewrite in exponential form.	$2^{4} = 3 x - 5$
Simplify.	$16 = 3 x - 5$
Solve for x.	$x = 7$
Check.	$\log_{2} (3 (7) - 5) = 4$
	$4 = 4$ , True

Example 4

Solve $\log_{4} (x + 6) - \log_{4} (2 x + 5) = - \log_{4} x$

Solution

.
Use the Quotient Property on the left side and the Power Property on the right.	$\log_{4} \frac{x + 6}{2 x + 5} = \log_{4} x^{- 1}$
Rewrite $x^{-1}$ as $\frac{1}{x}$ .	$\log_{4} \frac{x + 6}{2 x + 5} = \log_{4} \frac{1}{x}$
Use the One-to-One Property.	$\frac{x + 6}{2 x + 5} = \frac{1}{x}$
Solve the rational equation.	$x (x + 6) = 2 x + 5$
Distribute and write in standard form.	$x^{2} + 4 x - 5 = 0$
Factor and solve for x.	$(x + 5) (x - 1) = 0$ , $x = -5$ , $x = 1$
Check:	x=–5 is extraneous solution because $2 (-5) + 5 < 0$ so x=1 is the only solution.

Practice Makes Perfect

Don’t forget to check your solutions.

$\log_{3} x = 5$

$\log_{2} (x + 1) + \log_{2} (x - 1) = 3$

$\log (x - 2) - \log (4 x + 16) = \log \frac{1}{x}$

Using Like Bases to Solve Exponential Equations

The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers $b,$ $S,$ and $T,$ where $b > 0, b \neq 1,$ $b^{S} = b^{T}$ if and only if $S = T .$

In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.

For example, consider the equation $3^{4 x - 7} = \frac{3^{2 x}}{3} .$ To solve for $x,$ we use the division property of exponents to rewrite the right side so that both sides have the common base, $3.$ Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for $x$ :

\begin{array}{l} 3^{4 x - 7} & = \frac{3^{2 x}}{3} \\ 3^{4 x - 7} & = \frac{3^{2 x}}{3^{1}} & {Rewrite 3 as 3}^{1} . \\ 3^{4 x - 7} & = 3^{2 x - 1} & Use the division property of exponents . \\ 4 x - 7 & = 2 x - 1 & Apply the one-to-one property of exponents . \\ 2 x & = 6 & Subtract 2 x and add 7 to both sides . \\ x & = 3 & Divide by 2 . \end{array}

Example 5

Solving an Exponential Equation with a Common Base

Solve $2^{x - 1} = 2^{2 x - 4} .$

Solution

\begin{array}{l} 2^{x - 1} = 2^{2 x - 4} & The common base is 2. \\ x - 1 = 2 x - 4 \begin{matrix}  \end{matrix} & By the one-to-one property the exponents must be equal . \\ x = 3 & Solve for x . \end{array}

Rewriting Equations So All Powers Have the Same Base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation $256 = 4^{x - 5} .$ We can rewrite both sides of this equation as a power of $2.$ Then we apply the rules of exponents, along with the one-to-one property, to solve for $x :$

\begin{array}{l} 256 = 4^{x - 5} \\ 2^{8} = {(2^{2})}^{x - 5} & Rewrite each side as a power with base 2 . \\ 2^{8} = 2^{2 x - 10} & Use the one-to-one property of exponents . \\ 8 = 2 x - 10 \begin{matrix}  \end{matrix} & Apply the one-to-one property of exponents . \\ 18 = 2 x & Add 10 to both sides . \\ x = 9 & Divide by 2 . \end{array}

Example 6

Solving Equations by Rewriting Them to Have a Common Base

Solve $8^{x + 2} = 16^{x + 1} .$

Solution

\begin{array}{l} 8^{x + 2} = 16^{x + 1} \\ {(2^{3})}^{x + 2} = {(2^{4})}^{x + 1} & Write 8 and 16 as powers of 2. \\ 2^{3 x + 6} = 2^{4 x + 4} & To take a power of a power, multiply exponents . \\ 3 x + 6 = 4 x + 4 & Use the one-to-one property to set the exponents equal . \\ x = 2 & Solve for x . \end{array}

Example 7

Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base

Solve $2^{5 x} = \sqrt{2} .$

Solution

\begin{array}{l} 2^{5 x} = 2^{\frac{1}{2}} & Write the square root of  2 as a power of 2. \\ 5 x = \frac{1}{2} & Use the one-to-one property . \\ x = \frac{1}{10} & Solve for x . \end{array}

Example 8

Solving an Equation with Positive and Negative Powers

Solve $3^{x + 1} = −2.$

Solution

This equation has no solution. There is no real value of $x$ that will make the equation a true statement because any power of a positive number is positive.

Analysis

Figure 2 shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution.

Graph of 3^(x+1)=-2 and y=-2. The graph notes that they do not cross. — Figure 2

Solving Exponential Equations Using Logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since $\log (a) = \log (b)$ is equivalent to $a = b,$ we may apply logarithms with the same base on both sides of an exponential equation.

Example 9

Solving an Equation Containing Powers of Different Bases

Solve $5^{x + 2} = 4^{x} .$

Solution

\begin{array}{l} 5^{x + 2} = 4^{x} & There is no easy way to get the powers to have the same base . \\ \ln 5^{x + 2} = \ln 4^{x} & Take ln of both sides . \\ (x + 2) \ln 5 = x \ln 4 & Use laws of logs . \\ x \ln 5 + 2 \ln 5 = x \ln 4 & Use the distributive law . \\ x \ln 5 - x \ln 4 = - 2 \ln 5 & Get terms containing x on one side, terms without x on the other . \\ x (\ln 5 - \ln 4) = - 2 \ln 5 & On the left hand side, factor out an x . \\ x \ln (\frac{5}{4}) = \ln (\frac{1}{25}) \begin{matrix}  \end{matrix} & Use the laws of logs. \\ x = \frac{\ln (\frac{1}{25})}{\ln (\frac{5}{4})} & Divide by the coefficient of x . \end{array}

Equations Containing e

One common type of exponential equations are those with base $e .$ This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base $e$ on either side, we can use the natural logarithm to solve it.

Example 10

Solve an Equation of the Form y = Ae^kt

Solve $100 = 20 e^{2 t} .$

Solution

\begin{array}{l} 100 & = 20 e^{2 t} \\ 5 & = e^{2 t} & Divide by the coefficient of the power. \\ \ln 5 & = 2 t & Take ln of both sides . Use the fact that \ln (x) and e^{x} are inverse functions . \\ t & = \frac{\ln 5}{2} & Divide by the coefficient of t . \end{array}

Analysis

Using laws of logs, we can also write this answer in the form $t = \ln \sqrt{5} .$ If we want a decimal approximation of the answer, we use a calculator.

Example 11

Solving an Equation That Can Be Simplified to the Form y = Ae^kt

Solve $4 e^{2 x} + 5 = 12.$

Solution

\begin{array}{l} 4 e^{2 x} + 5 = 12 \\ 4 e^{2 x} = 7 & Combine like terms . \\ e^{2 x} = \frac{7}{4} & Divide by the coefficient of the power . \\ 2 x = \ln (\frac{7}{4}) & Take ln of both sides . \\ x = \frac{1}{2} \ln (\frac{7}{4}) & Solve for x . \end{array}

Extraneous Solutions

Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

Example 12

Solving Exponential Functions in Quadratic Form

Solve $e^{2 x} - e^{x} = 56.$

Solution

\begin{array}{l} e^{2 x} - e^{x} & = 56 \\ e^{2 x} - e^{x} - 56 & = 0 & Get one side of the equation equal to zero . \\ (e^{x} + 7) (e^{x} - 8) & = 0 & Factor by the FOIL method . \\ e^{x} + 7 & = 0 or e^{x} - 8 = 0 \begin{matrix}  \end{matrix} & If a product is zero, then one factor must be zero . \\ e^{x} & = - 7 {or e}^{x} = 8 & Isolate the exponentials . \\ e^{x} & = 8 & Reject the equation in which the power equals a negative number . \\ x & = \ln 8 & Solve the equation in which the power equals a positive number . \end{array}

Analysis

When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation $e^{x} = −7$ because a positive number never equals a negative number. The solution $\ln (−7)$ is not a real number, and in the real number system this solution is rejected as an extraneous solution.

Using the Definition of a Logarithm to Solve Logarithmic Equations

We have already seen that every logarithmic equation $\log_{b} (x) = y$ is equivalent to the exponential equation $b^{y} = x .$ We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.

For example, consider the equation $\log_{2} (2) + \log_{2} (3 x - 5) = 3.$ To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for $x :$

\begin{array}{l} \log_{2} (2) + \log_{2} (3 x - 5) = 3 \\ \log_{2} (2 (3 x - 5)) = 3 & Apply the product rule of logarithms. \\ \log_{2} (6 x - 10) = 3 & Distribute . \\ 2^{3} = 6 x - 10 & Apply the definition of a logarithm . \\ 8 = 6 x - 10 \begin{matrix}  \end{matrix} & Calculate 2^{3} . \\ 18 = 6 x & Add 10 to both sides . \\ x = 3 & Divide by 6 . \end{array}

Example 13

Using Algebra to Solve a Logarithmic Equation

Solve $2 \ln x + 3 = 7.$

Solution

\begin{array}{l} 2 \ln x + 3 = 7 \\ 2 \ln x = 4 & Subtract 3 . \\ \ln x = 2 & Divide by 2 . \\ x = e^{2} & Rewrite in exponential form . \end{array}

Example 14

Using Algebra Before and After Using the Definition of the Natural Logarithm

Solve $2 \ln (6 x) = 7.$

Solution

\begin{array}{l} 2 \ln (6 x) = 7 \\ \ln (6 x) = \frac{7}{2} & Divide by 2 . \\ 6 x = e^{(\frac{7}{2})} & Use the definition of \ln . \\ x = \frac{1}{6} e^{(\frac{7}{2})} & Divide by 6 . \end{array}

Example 15

Using a Graph to Understand the Solution to a Logarithmic Equation

Solve $\ln x = 3.$

Solution

\begin{array}{l} \ln x = 3 \\ x = e^{3} & Use the definition of the natural logarithm . \end{array}

Figure 3 represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20. In other words $e^{3} \approx 20.$ A calculator gives a better approximation: $e^{3} \approx 20.0855.$

Graph of two questions, y=3 and y=ln(x), which intersect at the point (e^3, 3) which is approximately (20.0855, 3). — Figure 3 The graphs of $y = \ln x$ and $y = 3$ cross at the point ${(e}^{3}, 3),$ which is approximately (20.0855, 3).

Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers $x > 0,$ $S > 0,$ $T > 0$ and any positive real number $b,$ where $b \neq 1,$

\log_{b} S = \log_{b} T if and only if S = T .

For example,

If \log_{2} (x - 1) = \log_{2} (8), then x - 1 = 8.

So, if $x - 1 = 8,$ then we can solve for $x,$ and we get $x = 9.$ To check, we can substitute $x = 9$ into the original equation: $\log_{2} (9 - 1) = \log_{2} (8) = 3.$ In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.

For example, consider the equation $\log (3 x - 2) - \log (2) = \log (x + 4) .$ To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for $x :$

\begin{array}{l} \log (3 x - 2) - \log (2) = \log (x + 4) \\ \log (\frac{3 x - 2}{2}) = \log (x + 4) & Apply the quotient rule of logarithms . \\ \frac{3 x - 2}{2} = x + 4 & Apply the one to one property of a logarithm . \\ 3 x - 2 = 2 x + 8 & Multiply both sides of the equation by 2. \\ x = 10 & Subtract 2 x and add 2 . \end{array}

To check the result, substitute $x = 10$ into $\log (3 x - 2) - \log (2) = \log (x + 4) .$

\begin{array}{l} \log (3 (10) - 2) - \log (2) = \log ((10) + 4) \\ \log (28) - \log (2) = \log (14) \\ \log (\frac{28}{2}) = \log (14) & The solution checks . \end{array}

Example 16

Solving an Equation Using the One-to-One Property of Logarithms

Solve $\ln (x^{2}) = \ln (2 x + 3) .$

Solution

\begin{array}{l} \ln (x^{2}) = \ln (2 x + 3) \\ x^{2} = 2 x + 3 & Use the one-to-one property of the logarithm . \\ x^{2} - 2 x - 3 = 0 & Get zero on one side before factoring . \\ (x - 3) (x + 1) = 0 & Factor using FOIL . \\ x - 3 = 0 or x + 1 = 0 \begin{matrix}  \end{matrix} & If a product is zero, one of the factors must be zero . \\ x = 3 or x = - 1 & Solve for x . \end{array}

Analysis

There are two solutions: $3$ or $−1.$ The solution $−1$ is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive.

Solving Applied Problems Using Exponential and Logarithmic Equations

In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.

One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. Table 1 lists the half-life for several of the more common radioactive substances.

Table 1 Seven rows and three columns. The first column is labeled, “substance”, the second column is labeled, “use”, and the third column is labeled, “half-life”. Gallium-67 is used for nuclear medicine and has a half-life of 80 hours. Cobalt-60 is used for manufacturing and has a half-life of 5.3 years. Technetium-99m is used for nuclear medicine and has a half-life of 6 hours. Americium-241 is used for construction and has a half-life of 432 years. Carbon-14 is used for archeological dating and has a half-life of 5,715 years. Uranium-235 is used for atomic power and has a half-life of 703,800,000 years.
Substance	Use	Half-life
gallium-67	nuclear medicine	80 hours
cobalt-60	manufacturing	5.3 years
technetium-99m	nuclear medicine	6 hours
americium-241	construction	432 years
carbon-14	archeological dating	5,730 years
uranium-235	atomic power	703,800,000 years

We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:

\begin{array}{l} A (t) = A_{0} e^{\frac{\ln (0.5)}{T} t} \\ A (t) = A_{0} e^{\ln (0.5) \frac{t}{T}} \\ A (t) = A_{0} {(e^{\ln (0.5)})}^{\frac{t}{T}} \\ A (t) = A_{0} {(\frac{1}{2})}^{\frac{t}{T}} \end{array}

where

$A_{0}$ is the amount initially present
$T$ is the half-life of the substance
$t$ is the time period over which the substance is studied
$A (t)$ is the amount of the substance present after time $t$

Example 17

Using the Formula for Radioactive Decay to Find the Quantity of a Substance

How long will it take for ten percent of a 1000-gram sample of uranium-235 to decay?

Solution

\begin{array}{l} y = 1000 e \frac{\ln (0.5)}{703,800,000} t \\ 900 = 1000 e^{\frac{\ln (0.5)}{703,800,000} t} & After 10% decays, 900 grams are left . \\ 0.9 = e^{\frac{\ln (0.5)}{703,800,000} t} & Divide by 1000 . \\ \ln (0.9) = \ln (e^{\frac{\ln (0.5)}{703,800,000} t}) & Take ln of both sides . \\ \ln (0.9) = \frac{\ln (0.5)}{703,800,000} t & ln (e^{M}) = M \\ t = 703,800,000 \times \frac{\ln (0.9)}{\ln (0.5)} years \begin{matrix}  \end{matrix} & Solve for t . \\ t \approx 106,979,777 years \end{array}

Analysis

Ten percent of 1000 grams is 100 grams. If 100 grams decay, the amount of uranium-235 remaining is 900 grams.

Key Equations

...
One-to-one property for exponential functions	For any algebraic expressions $S$ and $T$ and any positive real number $b,$ where $b^{S} = b^{T}$ if and only if $S = T .$
Definition of a logarithm	For any algebraic expression S and positive real numbers $b$ and $c,$ where $b \neq 1,$ $\log_{b} (S) = c$ if and only if $b^{c} = S .$
One-to-one property for logarithmic functions	For any algebraic expressions S and T and any positive real number $b,$ where $b \neq 1,$ $\log_{b} S = \log_{b} T$ if and only if $S = T .$

Key Concepts

We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See Example 5.
When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See Example 6, Example 7, and Example 8.
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See Example 9.
We can solve exponential equations with base $e,$ by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See Example 10 and Example 11.
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See Example 12.
When given an equation of the form $\log_{b} (S) = c,$ where $S$ is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation $b^{c} = S,$ and solve for the unknown. See Example 13 and Example 14.
We can also use graphing to solve equations with the form $\log_{b} (S) = c .$ We graph both equations $y = \log_{b} (S)$ and $y = c$ on the same coordinate plane and identify the solution as the x-value of the intersecting point. See Example 15.
When given an equation of the form $\log_{b} S = \log_{b} T,$ where $S$ and $T$ are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation $S = T$ for the unknown. See Example 16.
Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See Example 17.

Section Exercises

Verbal

Exercise 1

How can an exponential equation be solved?

Solution

Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.

Exercise 2

When does an extraneous solution occur? How can an extraneous solution be recognized?

Exercise 3

When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

Solution

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

Algebraic

For the following exercises, use like bases to solve the exponential equation.

Exercise 4

$4^{- 3 v - 2} = 4^{- v}$

Exercise 5

$64 \cdot 4^{3 x} = 16$

Solution

$x = - \frac{1}{3}$

Exercise 6

$3^{2 x + 1} \cdot 3^{x} = 243$

Exercise 7

$2^{- 3 n} \cdot \frac{1}{4} = 2^{n + 2}$

Solution

$n = - 1$

Exercise 8

$625 \cdot 5^{3 x + 3} = 125$

Exercise 9

$\frac{36^{3 b}}{36^{2 b}} = 216^{2 - b}$

Solution

$b = \frac{6}{5}$

Exercise 10

${(\frac{1}{64})}^{3 n} \cdot 8 = 2^{6}$

For the following exercises, use logarithms to solve.

Exercise 11

$9^{x - 10} = 1$

Solution

$x = 10$

Exercise 12

$2 e^{6 x} = 13$

Exercise 13

$e^{r + 10} - 10 = −42$

Solution

No solution

Exercise 14

$2 \cdot 10^{9 a} = 29$

Exercise 15

$- 8 \cdot 10^{p + 7} - 7 = −24$

Solution

$p = \log (\frac{17}{8}) - 7$

Exercise 16

$7 e^{3 n - 5} + 5 = −89$

Exercise 17

$e^{- 3 k} + 6 = 44$

Solution

$k = - \frac{\ln (38)}{3}$

Exercise 18

$- 5 e^{9 x - 8} - 8 = −62$

Exercise 19

$- 6 e^{9 x + 8} + 2 = −74$

Solution

$x = \frac{\ln (\frac{38}{3}) - 8}{9}$

Exercise 20

$2^{x + 1} = 5^{2 x - 1}$

Exercise 21

$e^{2 x} - e^{x} - 132 = 0$

Solution

$x = \ln 12$

Exercise 22

$7 e^{8 x + 8} - 5 = −95$

Exercise 23

$10 e^{8 x + 3} + 2 = 8$

Solution

$x = \frac{\ln (\frac{3}{5}) - 3}{8}$

Exercise 24

$4 e^{3 x + 3} - 7 = 53$

Exercise 25

$8 e^{- 5 x - 2} - 4 = −90$

Solution

no solution

Exercise 26

$3^{2 x + 1} = 7^{x - 2}$

Exercise 27

$e^{2 x} - e^{x} - 6 = 0$

Solution

$x = \ln (3)$

Exercise 28

$3 e^{3 - 3 x} + 6 = −31$

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

Exercise 29

$\log (\frac{1}{100}) = −2$

Solution

$10^{- 2} = \frac{1}{100}$

Exercise 30

$\log_{324} (18) = \frac{1}{2}$

For the following exercises, use the definition of a logarithm to solve the equation.

Exercise 31

$5 \log_{7} n = 10$

Solution

$n = 49$

Exercise 32

$- 8 \log_{9} x = 16$

Exercise 33

$4 + \log_{2} (9 k) = 2$

Solution

$k = \frac{1}{36}$

Exercise 34

$2 \log (8 n + 4) + 6 = 10$

Exercise 35

$10 - 4 \ln (9 - 8 x) = 6$

Solution

$x = \frac{9 - e}{8}$

For the following exercises, use the one-to-one property of logarithms to solve.

Exercise 36

$\ln (10 - 3 x) = \ln (- 4 x)$

Exercise 37

$\log_{13} (5 n - 2) = \log_{13} (8 - 5 n)$

Solution

$n = 1$

Exercise 38

$\log (x + 3) - \log (x) = \log (74)$

Exercise 39

$\ln (- 3 x) = \ln (x^{2} - 6 x)$

Solution

No solution

Exercise 40

$\log_{4} (6 - m) = \log_{4} 3 m$

Exercise 41

$\ln (x - 2) - \ln (x) = \ln (54)$

Solution

No solution

Exercise 42

$\log_{9} (2 n^{2} - 14 n) = \log_{9} (- 45 + n^{2})$

Exercise 43

$\ln (x^{2} - 10) + \ln (9) = \ln (10)$

Solution

$x = \pm \frac{10}{3}$

For the following exercises, solve each equation for $x .$

Exercise 44

$\log (x + 12) = \log (x) + \log (12)$

Exercise 45

$\ln (x) + \ln (x - 3) = \ln (7 x)$

Solution

$x = 10$

Exercise 46

$\log_{2} (7 x + 6) = 3$

Exercise 47

$\ln (7) + \ln (2 - 4 x^{2}) = \ln (14)$

Solution

$x = 0$

Exercise 48

$\log_{8} (x + 6) - \log_{8} (x) = \log_{8} (58)$

Exercise 49

$\ln (3) - \ln (3 - 3 x) = \ln (4)$

Solution

$x = \frac{3}{4}$

Exercise 50

$\log_{3} (3 x) - \log_{3} (6) = \log_{3} (77)$

Graphical

For the following exercises, solve the equation for $x,$ if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

Exercise 51

$\log_{9} (x) - 5 = −4$

Solution

$x = 9$

Exercise 52

$\log_{3} (x) + 3 = 2$

Exercise 53

$\ln (3 x) = 2$

Solution

$x = \frac{e^{2}}{3} \approx 2.5$

Exercise 54

$\ln (x - 5) = 1$

Exercise 55

$\log (4) + \log (- 5 x) = 2$

Solution

$x = - 5$

Exercise 56

$- 7 + \log_{3} (4 - x) = −6$

Exercise 57

$\ln (4 x - 10) - 6 = - 5$

Solution

$x = \frac{e + 10}{4} \approx 3.2$

Exercise 58

$\log (4 - 2 x) = \log (- 4 x)$

Exercise 59

$\log_{11} (- 2 x^{2} - 7 x) = \log_{11} (x - 2)$

Solution

No solution

Graph of log_11(-2x^2-7x)=y and y=log_11(x-2).

Exercise 60

$\ln (2 x + 9) = \ln (- 5 x)$

Exercise 61

$\log_{9} (3 - x) = \log_{9} (4 x - 8)$

Solution

$x = \frac{11}{5} \approx 2.2$

Graph of log_9(3-x)=y and y=log_9(4x-8).

Exercise 62

$\log (x^{2} + 13) = \log (7 x + 3)$

Exercise 63

$\frac{3}{\log_{2} (10)} - \log (x - 9) = \log (44)$

Solution

$x = \frac{101}{11} \approx 9.2$

Graph of 3/log_2(10)-log(x-9)=y and y=log(44).

Exercise 64

$\ln (x) - \ln (x + 3) = \ln (6)$

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

Exercise 65

An account with an initial deposit of $$6,500$ earns $7.25 %$ annual interest, compounded continuously. How much will the account be worth after 20 years?

Solution

about $$ 27, 710.24$

Graph of f(x)=6500e^(0.0725x) with the labeled point at (20, 27710.24).

Exercise 66

The formula for measuring sound intensity in decibels $D$ is defined by the equation $D = 10 \log (\frac{I}{I_{0}}),$ where $I$ is the intensity of the sound in watts per square meter and $I_{0} = 10^{- 12}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of $8.3 \cdot 10^{2}$ watts per square meter?

Exercise 67

The population of a small town is modeled by the equation $P = 1650 e^{0.5 t}$ where $t$ is measured in years. In approximately how many years will the town’s population reach $20,000?$

Solution

about 5 years

Graph of P(t)=1650e^(0.5x) with the labeled point at (5, 20000).

Technology

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

Exercise 68

$1000 {(1.03)}^{t} = 5000$ using the common log.

Exercise 69

$e^{5 x} = 17$ using the natural log

Solution

$\frac{\ln (17)}{5} \approx 0.567$

Exercise 70

$3 {(1.04)}^{3 t} = 8$ using the common log

Exercise 71

$3^{4 x - 5} = 38$ using the common log

Solution

$x = \frac{\log (38) + 5 \log (3)}{4 \log (3)} \approx 2.078$

Exercise 72

$50 e^{- 0.12 t} = 10$ using the natural log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

Exercise 73

$7 e^{3 x - 5} + 7.9 = 47$

Solution

$x \approx 2.2401$

Exercise 74

$\ln (3) + \ln (4.4 x + 6.8) = 2$

Exercise 75

$\log (- 0.7 x - 9) = 1 + 5 \log (5)$

Solution

$x \approx - 44655.7143$

Exercise 76

Atmospheric pressure $P$ in pounds per square inch is represented by the formula $P = 14.7 e^{- 0.21 x},$ where $x$ is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of $8.369$ pounds per square inch? (Hint: there are 5280 feet in a mile)

Exercise 77

The magnitude M of an earthquake is represented by the equation $M = \frac{2}{3} \log (\frac{E}{E_{0}})$ where $E$ is the amount of energy released by the earthquake in joules and $E_{0} = 10^{4.4}$ is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing $1.4 \cdot 10^{13}$ joules of energy?

Solution

about $5.83$

Extensions

Exercise 78

Use the definition of a logarithm along with the one-to-one property of logarithms to prove that $b^{\log_{b} x} = x .$

Exercise 79

Recall the formula for continually compounding interest, $y = A e^{k t} .$ Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.

Solution

$t = \ln ({(\frac{y}{A})}^{\frac{1}{k}})$

Exercise 80

Recall the compound interest formula $A = a {(1 + \frac{r}{k})}^{k t} .$ Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t .$

Exercise 81

Newton’s Law of Cooling states that the temperature $T$ of an object at any time t can be described by the equation $T = T_{s} + (T_{0} - T_{s}) e^{- k t},$ where $T_{s}$ is the temperature of the surrounding environment, $T_{0}$ is the initial temperature of the object, and $k$ is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.

Solution

$t = \ln ({(\frac{T - T_{s}}{T_{0} - T_{s}})}^{- \frac{1}{k}})$

Exponential and Logarithmic Equations

Learning Objectives

Objective 1: Solve Exponential Equations. (IA 10.2.2)

Solving Exponential Equations.

Practice Makes Perfect

Objective 2: Solving Logarithmic Equations. (IA 10.3.4)

Solving logarithmic equations.

Practice Makes Perfect

Using Like Bases to Solve Exponential Equations

Solving an Exponential Equation with a Common Base

Rewriting Equations So All Powers Have the Same Base

Solving Equations by Rewriting Them to Have a Common Base

Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base

Solving an Equation with Positive and Negative Powers

Analysis

Solving Exponential Equations Using Logarithms

Solving an Equation Containing Powers of Different Bases

Equations Containing e

Solve an Equation of the Form y = Aekt

Analysis

Solving an Equation That Can Be Simplified to the Form y = Aekt

Extraneous Solutions

Solving Exponential Functions in Quadratic Form

Analysis

Using the Definition of a Logarithm to Solve Logarithmic Equations

Using Algebra to Solve a Logarithmic Equation

Using Algebra Before and After Using the Definition of the Natural Logarithm

Using a Graph to Understand the Solution to a Logarithmic Equation

Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

Solving an Equation Using the One-to-One Property of Logarithms

Analysis

Solving Applied Problems Using Exponential and Logarithmic Equations

Using the Formula for Radioactive Decay to Find the Quantity of a Substance

Analysis

Key Equations

Key Concepts

Section Exercises

Verbal

Algebraic

Graphical

Technology

Extensions

Solve an Equation of the Form y = Ae^kt

Solving an Equation That Can Be Simplified to the Form y = Ae^kt