Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

Solving logarithmic and exponential equations

Last updated

Feb 23, 2023
Save as PDF
- 4.7: Chapter Review
- 5: Introduction to Calculus

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Learning Objectives

Use like bases to solve exponential equations.
Use logarithms to solve exponential equations.
Use the definition of a logarithm to solve logarithmic equations.
Use the one-to-one property of logarithms to solve logarithmic equations.
Solve applied problems involving exponential and logarithmic equations

In 1859, an Australian landowner named Thomas Austin released 2424 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.

Figure 4.6.14.6.1: Wild rabbits in Australia. The rabbit population grew so quickly in Australia that the event became known as the “rabbit plague.” (credit: Richard Taylor, Flickr)

Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

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, S>0, b≠1, b ≠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

${3}^{4x-7} = \frac{3^{2x}}{3}$

${3}^{4x-7} = \frac{3^{2x}}{3^{1}}$ Rewrite 3 as {3}^{1}

${3}^{4x-7} = {3}^{2x-1}$ Use the division property of exponents

$4x-7 = 2x - 1$ The bases are the same, so the exponents are too.

$2x = 6$ Subtract 2x and add 7 to both sides

$x = 3$ Divide both sides by 2. To solve for x�, we use the division property of exponents to rewrite the right side so that both sides have the common base, 33. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for x�:

USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS

For any algebraic expressions S and T, and any positive real number

$b\neq 1$ .

b^S = b^T if and only if S = T

How to: Given an exponential equation with the form b^S = b^T, where S and T are algebraic expressions with an unknown, solve for the unknown.

Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form b^S = b^T.
Use the one-to-one property to set the exponents equal.
Solve the resulting equation, S = T, for the unknown.

Example $\PageIndex{1}$ Solving an Exponential Equation with a Common Base

Solve 2^x-1 = 2^2x^-4.

Solution

2^x-1 = 2^2x^-4 The common base is 2

x - 1 = 2x - 4 By the one-to-one property the exponents must be equal

x = 3 Solve for x

Example $\PageIndex{2}$

Solve 5^2x= 5^3x⁺²

Solution

2x = 3x + 2 subtract 3x from both sides
-3x -3x
-x = 0 + 2
-x = 2 divide both sides by -1
x = -2

Rewrite 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.

How to: Given an exponential equation with unlike bases, use the one-to-one property to solve it.

Rewrite each side in the equation as a power with a common base.
Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form b^S=b^T.
Use the one-to-one property to set the exponents equal.
Solve the resulting equation, S=T, for the unknown.

Example $\PageIndex{3}$

Solving Equations by Rewriting Them to Have a Common Base

Add example text here.

Solution

Add example text here.

For example, consider the equation 256=4^x−5. We can rewrite both sides of this equation as a power of 22. Then we apply the rules of exponents, along with the one-to-one property, to solve for x:

$256 = 4^{x-5}$

$2^{8} = (2^{2})^{x-5}$ Rewrite each side as a power with base

$2^{8} = 2^{2x-10}$ Use the one-to-one property of exponents

$8 = 2x -10$ Apply to one-to-one property of exponents

$18 = 2x$ Add 10 to both sides

$x = 9$ Divide both sides by 2

Search

Text Color

Text Size

Margin Size

Font Type

USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS

How to: Given an exponential equation with the form b^S = b^T, where S and T are algebraic expressions with an unknown, solve for the unknown.

Example $\PageIndex{1}$ Solving an Exponential Equation with a Common Base

Example $\PageIndex{2}$

Solution

How to: Given an exponential equation with unlike bases, use the one-to-one property to solve it.

Example \(\PageIndex{3}\)

Solution

Learning Objectives

Using Like Bases to Solve Exponential Equations

USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS

How to: Given an exponential equation with the form bS = bT, where S and T are algebraic expressions with an unknown, solve for the unknown.

Example 1\PageIndex{1} Solving an Exponential Equation with a Common Base

Example 2\PageIndex{2}

Solution

Rewrite Equations So All Powers Have the Same Base

How to: Given an exponential equation with unlike bases, use the one-to-one property to solve it.

Example \(\PageIndex{3}\)

Solution

Support Center

How can we help?

How to: Given an exponential equation with the form b^S = b^T, where S and T are algebraic expressions with an unknown, solve for the unknown.

Example $\PageIndex{1}$ Solving an Exponential Equation with a Common Base

Example $\PageIndex{2}$