3.3.3: Inverse Properties of Logarithms
- Page ID
- 14374
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Inverse Properties of Logarithmic Functions
If you continue to study mathematics into college, you may take a course called Differential Equations. There you will learn that the solution to the differential equation y′=y is the general function y=Cex. What is the inverse of this function?
Inverse Properties of Logarithms
By the definition of a logarithm, it is the inverse of an exponent. Therefore, a logarithmic function is the inverse of an exponential function. Recall what it means to be an inverse of a function. When two inverses are composed, they equal \(\ x\). Therefore, if \(\ f(x)=b^{x} \text { and } g(x)=\log _{b} x\), then:
\(\ f \circ g=b^{\log _{b} x}=x \text { and } g \circ f=\log _{b} b^{x}=x\)
These are called the Inverse Properties of Logarithms.
Let's solve the following problems. We will use the Inverse Properties of Logarithms.
- Find \(\ 10^{\log 56}\).
Using the first property, we see that the bases cancel each other out. \(\ 10^{\log 56}=56\)
\(\ e^{\ln 6} \cdot e^{\ln 2}\)
Here, \(\ e\) and the natural log cancel out and we are left with 6⋅2=12.
- Find \(\ \log _{4} 16^{x}\)
We will use the second property here. Also, rewrite 16 as 42.
\(\ \log _{4} 16^{x}=\log _{4}\left(4^{2}\right)^{x}=\log _{4} 4^{2 x}=2 x\)
- Find the inverse of \(\ f(x)=2 e^{x-1}\).
Change \(\ f(x)\) to \(\ y\). Then, switch \(\ x\) and \(\ y\).
\(\ \begin{array}{l}
y=2 e^{x-1} \\
x=2 e^{y-1}
\end{array}\)Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.
\(\ \begin{array}{l}
\frac{x}{2}=e^{y-1} \\
\ln \left(\frac{x}{2}\right)=\ln e^{y-1}
\end{array}\)Recall the Inverse Properties of Logarithms from earlier in this concept. \(\ \log _{b} b^{x}=x\); applying this to the right side of our equation, we have \(\ \ln e^{y-1}=y-1\). Solve for \(\ y\).
\(\ \begin{array}{l}
\ln \left(\frac{x}{2}\right)=y-1 \\
\ln \left(\frac{x}{2}\right)+1=y
\end{array}\)Therefore, \(\ \ln \left(\frac{x}{2}\right)+1\) is the inverse of \(\ 2 e^{y-1}\).
Examples
Earlier, you were asked to find the inverse of \(\ y=C e^{x}\).
Solution
Switch x and y in the function \(\ y=C e^{x}\) and then solve for y.
\(\ \begin{array}{r}
x=C e^{y} \\
\frac{x}{C}=e^{y} \\
\ln \frac{x}{C}=\ln \left(e^{y}\right) \\
\ln \frac{x}{C}=y
\end{array}\)
Therefore, the inverse of \(\ y=C e^{x} \text { is } y=\ln \frac{x}{C}\).
Simplify \(\ 5^{\log _{5} 6 x}\).
Solution
Using the first inverse property, the log and the base cancel out, leaving \(\ 6x\) as the answer.
\(\ 5^{\log _{5} 6 x}=6 x\)
Simplify \(\ \log _{9} 81^{x+2}\).
Solution
Using the second inverse property and changing 81 into 92 we have:
\(\ \begin{aligned}
\log _{9} 81^{x+2} &=\log _{9} 9^{2(x+2)} \\
&=2(x+2) \\
&=2 x+4
\end{aligned}\)
Find the inverse of \(\ f(x)=4^{x+2}-5\).
Solution
\(\ \begin{aligned}
f(x) &=4^{x+2}-5 \\
y &=4^{x+2}-5 \\
x &=4^{y+2}-5 \\
x+5 &=4^{y+2} \\
\log _{4}(x+5) &=y+2 \\
\log _{4}(x+5)-2 &=y
\end{aligned}\)
Review
Use the Inverse Properties of Logarithms to simplify the following expressions.
- \(\ \log _{3} 27^{x}\)
- \(\ \log _{5}\left(\frac{1}{5}\right)^{x}\)
- \(\ \log _{2}\left(\frac{1}{32}\right)^{x}\)
- \(\ 10^{\log (x+3)}\)
- \(\ \log _{6} 36^{(x-1)}\)
- \(\ 9^{\log _{9}(3 x)}\)
- \(\ e^{\ln (x-7)}\)
- \(\ \log \left(\frac{1}{100}\right)^{3 x}\)
- \(\ \ln e^{(5 x-3)}\)
Find the inverse of each of the following exponential functions.
- \(\ y=3 e^{x+2}\)
- \(\ f(x)=\frac{1}{5} e^{\frac{x}{7}}\)
- \(\ y=2+e^{2 x-3}\)
- \(\ f(x)=7^{\frac{3}{x}+1-5}\)
- \(\ y=2(6)^{\frac{x-5}{2}}\)
- \(\ f(x)=\frac{1}{3}(8)^{\frac{x}{2}-5}\)
Vocabulary
| Term | Definition |
|---|---|
| Inverse Properties of Logarithms | The inverse properties of logarithms are \(\ \log _{b} b^{x}=x \text { and } b^{\log _{b} x}=x, b \neq 1\). |