AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsLogarithmic FunctionsLogarithmic Function Basics

Logarithmic Function Basics (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

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
- $f (x) = a \log_{b} x$
  - where $b$ is the base, with $b > 0$ and $b \neq 1$
  - and $a$ is a nonzero constant that vertically scales the function ( $a \neq 0$ )

When $a = 1$ , this simplifies to the basic logarithmic function
- $f (x) = \log_{b} x$
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 $x > 0$
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 $f (x) = \log_{b} x$ passes through the point $(1, 0)$
- 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 $\ln x$ , 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 $\log_{e} x$
E.g. $\ln e = \log_{e} e = 1$ (because $e^{1} = e$ )
- $\ln 1 = \log_{e} 1 = 0$ (because $e^{0} = 1$ )
- $\ln (e^{3}) = 3$ (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 $e$

Examiner Tips and Tricks

Make sure you are comfortable with the shorthand notation for logarithms.

$\ln x$ always means $\log_{e} x$
- while $\log x$ (with no base written) is used for $\log_{10} x$
Confusing these is a common error

Your calculator has dedicated buttons for both.

Typically "ln" for the natural logarithm and "log" for the common (base 10) logarithm

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