Logarithmic Function Basics (College Board AP® Precalculus): Study Guide

Logarithmic functions

What is the general form of a logarithmic function?

• In the preceding study guide, you learned about logarithmic expressions like ${\log }_{b}c$

  • A logarithmic function takes this idea and lets the input vary

• The general form of a logarithmic function is

  • $f\left(x\right)=a{\log }_{b}x$

    • where $b$ is the base, with $b>0$ and $b\ne 1$

    • and $a$ is a nonzero constant that vertically scales the function ($a\ne 0$)

• When $\mathit{a}\mathbf{=}\mathbf{1}$, this simplifies to the basic logarithmic function

  • $f\left(x\right)={\log }_{b}x$

• The input $x$ must be positive (you cannot take the logarithm of zero or a negative number)

  • so the domain of a logarithmic function in general form is $\mathit{x}\mathbf{>}\mathbf{0}$

• Logarithmic functions are closely related to exponential functions

  • ${\log }_{b}x$ and $b^x$ are inverse functions of each other

  • This relationship is explored in more detail in the study guides that follow

How does the base affect a logarithmic function?

• The base $b$ determines how quickly the logarithmic function grows

  • A larger base means the function grows more slowly

    • because a larger base needs a smaller exponent to reach the same value

• E.g. ${\log }_{10}100=2$, but ${\log }_{2}100\approx 6.644$

  • the base-2 logarithm gives a larger output because base 2 needs more multiplications to reach 100

• Regardless of the base, every logarithmic function of the form $f\left(x\right)={\log }_{b}x$ passes through the point $\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}$

  • because ${\log }_{b}1=0$ for any valid base (since $b^0=1$)

The natural logarithm ln

What is the natural logarithm?

• The natural logarithm, written $\mathbf{ln}\mathit{x}$, is a logarithmic function with the natural base $e$

  • Recall from from the General Exponential Functions study guide that $e\approx 2.718$ is a special mathematical constant used frequently in modeling

• In other words, $\ln x$ is just shorthand for ${\mathbf{log}}_{\mathit{e}}\mathit{x}$

• E.g. $\ln e={\log }_{e}e=1$ (because $e^1=e$)

  • $\ln 1={\log }_{e}1=0$ (because $e^0=1$)

  • $\ln \left(e^3\right)=3$ (because $e^3=e^3$)

• The natural logarithm follows all the same rules as any other logarithm

  • The only difference is that the base is $e$ instead of some other number

• You will encounter $\ln$ frequently in this course, particularly when working with exponential models that use base $e$