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3.3.3: Inverse Properties of Logarithms

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    14374
  • 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.

    1. 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.

    2. Find \(\ \log _{4} 16^{x}\)

      We will use the second property here. Also, rewrite 16 as 42.

      \(\ \log _{4} 16^{x}=\log _{4}\left(4^{2}\right)^{x}=\log _{4} 4^{2 x}=2 x\)

    3. 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

    Example 1

    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 \(\ y=C e^{x} \text { is } y=\ln \frac{x}{C}\).

    Example 2

    Simplify \(\ 5^{\log _{5} 6 x}\).

    Solution

    Using the first inverse property, the log and the base cancel out, leaving \(\ 6x\) as the answer. 

    \(\ 5^{\log _{5} 6 x}=6 x\)

    Example 3

    Simplify \(\ \log _{9} 81^{x+2}\).

    Solution

    Using the second inverse property and changing 81 into 92 we have:

    \(\ \begin{aligned}
    \log _{9} 81^{x+2} &=\log _{9} 9^{2(x+2)} \\
    &=2(x+2) \\
    &=2 x+4
    \end{aligned}\)

    Example 4

    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.

    1. \(\ \log _{3} 27^{x}\)
    2. \(\ \log _{5}\left(\frac{1}{5}\right)^{x}\)
    3. \(\ \log _{2}\left(\frac{1}{32}\right)^{x}\)
    4. \(\ 10^{\log (x+3)}\)
    5. \(\ \log _{6} 36^{(x-1)}\)
    6. \(\ 9^{\log _{9}(3 x)}\)
    7. \(\ e^{\ln (x-7)}\)
    8. \(\ \log \left(\frac{1}{100}\right)^{3 x}\)
    9. \(\ \ln e^{(5 x-3)}\)

    Find the inverse of each of the following exponential functions.

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

    Answers for Review Problems

    To see the Review answers, open this PDF file and look for section 8.6. 


    Vocabulary

    Term Definition
    Inverse Properties of Logarithms The inverse properties of logarithms are \(\ \log _{b} b^{x}=x \text { and } b^{\log _{b} x}=x, b \neq 1\).
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