Logarithmic Functions

Logarithmic functions are fundamental in AP Precalculus because they:

• are the inverse functions of exponential functions

• allow us to solve exponential equations

• model quantities that grow or change by orders of magnitude

• connect algebraic structure with graphical behavior

Definition of a Logarithmic Function

A logarithmic function has the form:

$$f(x)=\log_{a}x$$

where:

• a>0, a≠1 is the base

• x>0 is the argument

This definition comes from the inverse relationship:

$$\log_{a}x=y \Longleftrightarrow a^y=x$$

Domain and Range

Domain

x>0

Logarithmic functions are undefined for zero or negative inputs.

Range

(−∞,∞)

Key Graphical Features

For $$f(x)=\log_{a}x$$:

• Vertical asymptote: x=0

• x-intercept: (1,0)

• No y-intercept

• Increasing if a>1

• Decreasing if 0<a<1

The graph is a reflection of $$y=a^x$$ across y=x.

Transformations of Logarithmic Functions

General form:

$$f(x)=a\log_{b}(x-h)+k$$

Effects:

• a: vertical stretch or reflection

• h: horizontal shift (affects domain)

• k: vertical shift

Example:

$$f(x)=\log(x-3)+2$$

• Domain: x>3

• Asymptote: x=3