# Logarithmic Functions

## Overview

### Description

Solving the equation $2^x = 8$ involves finding the value of the exponent $x$ to which the base 2 must be raised to equal 8. The equation, and others like it, can be solved by finding a logarithm. A logarithm is the exponent $x$ to which a base $b$ is raised to produce a given value $y$. In general, the exponential form $y=b^x$ is equivalent to the logarithmic form $x=\log_{b}y$. So, $2^{3} = 8$ is equivalent to $\log_{2}8=3$.

The function $f(x)=\log_b{x}$, where $b>0$ and $b\neq1$, is the parent logarithmic function. The graph of a logarithmic function is a curve that is either strictly increasing or strictly decreasing. The graph also has a vertical asymptote, a vertical line that the curve approaches, but never reaches. Transformations of the parent function $f(x)=\log_b{x}$ can be used to graph other logarithmic functions.

### At A Glance

• Finding a logarithm means finding the exponent to which a base must be raised to equal a given value.
• A logarithmic function is the inverse of an exponential function.
• Properties of logarithms can be used to simplify logarithmic expressions.
• To determine the logarithm of a product, add the logarithms of the factors.
• To calculate the logarithm of a quotient, subtract the logarithm of the divisor from the logarithm of the dividend.
• To determine the logarithm of a power, multiply the exponent by the logarithm of the base.
• A logarithm can be changed from one base to another by using the formula $\log_b{x}=\frac{\log_a{x}}{\log_a{b}}$.
• The graph of a logarithmic function has a vertical asymptote and is defined on only one side of the asymptote.
• Logarithmic functions can be evaluated to plot points on a graph.
• For a logarithmic function of the form $f(x)=\log_{b}x$, the value of b determines whether the function is increasing or decreasing.
• Transformations can be used to graph logarithmic functions.