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8.6: Inverse Properties of Logarithmic Functions

Difficulty Level: At Grade Created by: CK-12
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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 \begin{align*}y' = y\end{align*} is the general function \begin{align*}y = Ce^x\end{align*}. What is the inverse of this function?

Guidance

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 (see the Inverse of a Function concept), they equal \begin{align*}x\end{align*}. Therefore, if \begin{align*}f(x)=b^x\end{align*} and \begin{align*}g(x)=\log_b x\end{align*}, then:

\begin{align*}f \circ g=b^{\log_b x}=x\end{align*} and \begin{align*}g \circ f =\log_b b^x=x\end{align*}

These are called the Inverse Properties of Logarithms.

Example A

Find:

a) \begin{align*}10^{\log 56}\end{align*}

b) \begin{align*}e^{\ln6} \cdot e^{\ln2}\end{align*}

Solution: For each of these examples, we will use the Inverse Properties.

a) Using the first property, we see that the bases cancel each other out. \begin{align*}10^{\log 56}=56\end{align*}

b) Here, \begin{align*}e\end{align*} and the natural log cancel out and we are left with \begin{align*}6 \cdot 2=12\end{align*}.

Example B

Find \begin{align*}\log_4 16^x\end{align*}

Solution: We will use the second property here. Also, rewrite 16 as \begin{align*}4^2\end{align*}.

\begin{align*}\log_4 16^x=\log_4 (4^2)^x=\log_4 4^{2x}=2x\end{align*}

Example C

Find the inverse of \begin{align*}f(x)=2e^{x-1}\end{align*}.

Solution: See the Finding the Inverse concept for the steps on how to find the inverse.

Change \begin{align*}f(x)\end{align*} to \begin{align*}y\end{align*}. Then, switch \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

\begin{align*}& y=2e^{x-1} \\ & x=2e^{y-1}\end{align*}

Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.

\begin{align*}& \frac{x}{2}=e^{y-1} \\ & \ln \left(\frac{x}{2}\right)= \ln e^{y-1}\end{align*}

Recall the Inverse Properties from earlier in this concept. \begin{align*}\log_b b^x=x\end{align*}; applying this to the right side of our equation, we have \begin{align*}\ln e^{y-1}=y-1\end{align*}. Solve for \begin{align*}y\end{align*}.

\begin{align*}& \ln \left(\frac{x}{2}\right)=y-1 \\ & \ln \left(\frac{x}{2}\right)+1=y\end{align*}

Therefore, \begin{align*}\ln \left(\frac{x}{2}\right)+1\end{align*} is the inverse of \begin{align*}2e^{y-1}\end{align*}.

Intro Problem Revisit Switch x and y in the function \begin{align*}y = Ce^x\end{align*} and then solve for y.

\begin{align*}x = Ce^y\\ \frac{x}{C} = e^y\\ ln \frac{x}{C} = ln (e^y)\\ ln \frac{x}{C} = y\end{align*}

Therefore, the inverse of \begin{align*}y = Ce^x\end{align*} is \begin{align*}y = ln \frac{x}{C}\end{align*}.

Guided Practice

1. Simplify \begin{align*}5^{\log_5 6x}\end{align*}.

2. Simplify \begin{align*}\log_9 81^{x+2}\end{align*}.

3. Find the inverse of \begin{align*}f(x)=4^{x+2}-5\end{align*}.

Answers

1. Using the first inverse property, the log and the base cancel out, leaving \begin{align*}6x\end{align*} as the answer.

\begin{align*}5^{\log_5 6x}=6x\end{align*}

2. Using the second inverse property and changing 81 into \begin{align*}9^2\end{align*} we have:

\begin{align*}\log_9 81^{x+2} &= \log_9 9^{2(x+2)} \\ &= 2(x+2) \\ &= 2x+4\end{align*}

3. Follow the steps from Example C to find the inverse.

\begin{align*}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{align*}

Vocabulary

Inverse Properties of Logarithms
\begin{align*}\log_b b^x=x\end{align*} and \begin{align*}b^{\log_b x}=x, b \ne 1\end{align*}

Practice

Use the Inverse Properties of Logarithms to simplify the following expressions.

  1. \begin{align*}\log_3 27^x\end{align*}
  2. \begin{align*}\log_5 \left(\frac{1}{5}\right)^x\end{align*}
  3. \begin{align*}\log_2 \left(\frac{1}{32}\right)^x\end{align*}
  4. \begin{align*}10^{\log(x+3)}\end{align*}
  5. \begin{align*}\log_6 36^{(x-1)}\end{align*}
  6. \begin{align*}9^{\log_9(3x)}\end{align*}
  7. \begin{align*}e^{\ln(x-7)}\end{align*}
  8. \begin{align*}\log \left(\frac{1}{100}\right)^{3x}\end{align*}
  9. \begin{align*}\ln e^{(5x-3)}\end{align*}

Find the inverse of each of the following exponential functions.

  1. \begin{align*}y=3e^{x+2}\end{align*}
  2. \begin{align*}f(x)=\frac{1}{5}e^\frac{x}{7}\end{align*}
  3. \begin{align*}y=2+e^{2x-3}\end{align*}
  4. \begin{align*}f(x)=7^{\frac{3}{x}+1-5}\end{align*}
  5. \begin{align*}y=2(6)^\frac{x-5}{2}\end{align*}
  6. \begin{align*}f(x)=\frac{1}{3}(8)^{\frac{x}{2}-5}\end{align*}

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Vocabulary

Inverse Properties of Logarithms

The inverse properties of logarithms are \log_b b^x=x and b^{\log_b x}=x, b \ne 1.

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