Logarithmic Functions as Inverses of Exponential Functions (College Board AP® Precalculus): Study Guide

Logarithmic functions as inverses of exponential functions

How are exponential and logarithmic functions related?

• The logarithmic function $f\left(x\right)={\log }_{b}x$

  • and the exponential function $g\left(x\right)=b^x$

    • where $b>0$ and $b\ne 1$

  • are inverse functions of each other

• This means they "undo" each other:

  • ${\mathbf{log}}_{\mathit{b}}\mathbf{\left(}{\mathit{b}}^{\mathit{x}}\mathbf{\right)}\mathbf{=}\mathit{x}$

    • taking the log of an exponential with the same base returns the exponent

  • ${\mathit{b}}^{\mathbf{l}\mathbf{o}{\mathbf{g}}_{\mathit{b}}\mathit{x}}\mathbf{=}\mathit{x}$

    • raising a base to a logarithm with the same base returns the original value inside the logarithm

• More formally, if $f\left(x\right)={\log }_{b}x$ and $g\left(x\right)=b^x$

  • then $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)=x$

How do the input-output behaviors of logarithmic and exponential functions compare?

• Exponential and logarithmic functions handle change in opposite ways

• In an exponential function

  • when input values increase by equal amounts (additively)

  • the output values change by equal ratios (multiplicatively)

• In a logarithmic function, the relationship is reversed

  • when input values change by equal ratios (multiplicatively)

  • the output values increase by equal amounts (additively)

• E.g. for $g\left(x\right)=2^x$

  • increasing $x$ by 1 (additive change)

  • always doubles the output (multiplicative change)

• For $f\left(x\right)={\log }_{2}x$

  • doubling $x$ (multiplicative change)

  • always increases the output by 1 (additive change)

• Notice that the tables are mirrors of each other

  • The inputs and outputs are swapped

  • This is exactly the inverse relationship at work

How do ordered pairs relate between the two functions?

• If $\mathbf{(}\mathit{s}\mathbf{,}\mathit{t}\mathbf{)}$ is a point on the graph of the exponential function $g\left(x\right)=b^x$

  • then $\mathbf{(}\mathit{t}\mathbf{,}\mathit{s}\mathbf{)}$ is a point on the graph of the logarithmic function $f\left(x\right)={\log }_{b}x$

• This follows directly from the inverse relationship

  • if $b^s=t$, then ${\log }_{b}t=s$

• E.g. since $2^3=8$

  • the point $(3,8)$ is on the graph of $g\left(x\right)=2^x$

  • and the point $(8,3)$ is on the graph of $f\left(x\right)={\log }_{2}x$

Graphs of logarithmic functions

How is the graph of a logarithmic function related to the graph of an exponential function?

• The graph of $f\left(x\right)={\log }_{b}x$ is the reflection of the graph of $g\left(x\right)=b^x$ over the line $\mathit{y}\mathbf{=}\mathit{x}$

  • This is true for any base $b$ (with $b>0$ and $b\ne 1$)

  • It follows from the general principle that the graph of any function and its inverse are reflections of each other over $y=x$

• Note how features of the graphs swap under reflection:

  • The exponential function has a horizontal asymptote ($y=0$)

    • the logarithmic function has a vertical asymptote ($x=0$)

  • The exponential function passes through $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}$

    • the logarithmic function passes through $\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}$

  • The exponential function has domain all real numbers and range $y>0$

    • the logarithmic function has domain $x>0$ and range all real numbers

  • These graph features are true for any base $b$ (with $b>0$ and $b\ne 1$)

Rewriting exponential functions using logarithms

How can logarithms be used to rewrite exponential expressions?

• Logarithms provide a way to change the base of an exponential expression

  • This can be useful for comparing or simplifying expressions

• The key identity is $b^x=c^{\left({\log }_{c}b\right)\left(x\right)}$

  • This works because $b=c^{{\log }_{c}b}$

    • I.e., by the inverse properties of logarithms and exponentials

    • $c^{\square }$ 'cancels' ${\log }_c$

  • Therefore $b^x={\left(c^{{\log }_{c}b}\right)}^x=c^{\left({\log }_{c}b\right)\left(x\right)}$

• This means any exponential function can be rewritten using any base you choose

  • With the notation used above, you can use the identity to rewrite an exponential with base $b$ as an exponential with base $c$

• E.g. rewrite $3^x$ using the base $e$

  • Use the identity

    • $3^x=e^{\left(\ln 3\right)\left(x\right)}$

      • since $\ln 3={\log }_{e}3$

  • Or with the brackets expanded

    • $3^x=e^{x\ln 3}$

• E.g. rewrite $5^x$ using base 10

  • Using the identity

    • $5^x={10}^{\left({\log }_{10}5\right)\left(x\right)}$

  • Or with the brackets expanded

    • $5^x={10}^{x\log 5}$