Topic: Exponential and Logarithmic Functions

Subtopic: Logarithmic Functions & Graphs

**Overview**

After studying exponential functions, it seems a natural question to ask, what is the inverse of an exponential function? What function "undoes" say 2^{x} enabling you to solve for x in the equation 2^{x}=5 for instance? Well, a __logarithmic function__ is the inverse of an exponential function. Logarithms such as log_{3}x (read "log base 3 of x") have their own graphs and algebraic rules. These special functions are the focus of this lesson.

The conversion from exponential to logarithmic form (and visa versa) is particularly important to learn. Try to keep in mind that 2^{3}=8 and log_{2}8=3 say the same thing just in different forms (exponential form and logarithmic form, respectively).

As you study these new functions be sure to examine them both algebraically and graphically. Recognize connections between logarithmic and exponential functions and the differences between a __common logarithm__ (log(x)) and a __natural logarithm__ (ln(x)).

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 13.2.1 Convert between exponential form and logarithmic form
- 13.2.2 Understand the differences and similarities among common logarithms, natural logarithms, and other based logarithms
- 13.2.3 Evaluate logarithmic functions algebraically and electronically
- 13.2.4 Graph logarithmic functions manually and electronically
- 13.2.5 For the basic logarithmic function f(x)=log
_{b}(x), know the basic shape of its graph, intercept points, asymptote lines, domain, and range - 13.2.6 Know the algebraic and graphical relationships between each of these pairs of functions: f(x)=b
^{x}and f^{-1}(x)=log_{b}(x), g(x)=10^{x}and g^{-1}(x)=log(x), h(x)=e^{x}and h^{-1}(x)=ln(x) - 13.2.7 Recognize and be able to perform reflections (flips across the x-axis), stretches, vertical shifts (up or down), and horizontal shifts (left or right) in regards to logarithmic functions
- 13.2.8 Know what values in a logarithmic function's equation controls the above features (shifts, etc.) of the graph

**Terminology**

Terms you should be able to define: base of logarithm, input to logarithm, output from logarithm, notation associated with writing logarithms, common logarithm, natural logarithm, logarithmic function, vertical asymptote line