Algebra I / Elem. Algebra
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Factoring Rational Expressions Rational Equations and Applications
Algebra III / Inter. Algebra
Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
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S = contains supplemental resources
Course: Algebra III / Intermediate Algebra
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 2x enabling you to solve for x in the equation 2x=5 for instance? Well, a logarithmic function is the inverse of an exponential function. Logarithms such as log3x (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 23=8 and log28=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)=logb(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)=bx and f-1(x)=logb(x), g(x)=10x and g-1(x)=log(x), h(x)=ex 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

Define: base of logarithm, input to logarithm, output from logarithm, notation associated with writing logarithms, common logarithm, natural logarithm, logarithmic function, vertical asymptote line