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. These special functions are the focus of this lesson. Logarithms such as log3x (read "log base 3 of x") have their own graphs and algebraic rules. Be sure to memorize the properties of logarithms and the rules of logarithms which define the operations that can be conducted on logarithms as we enter a whole new algebra!
One caution: These rules of logarithms show that, for instance, log(2x) does not equal log(2)*log(x) and log(x+2) does not equal log(x)+log(2). Be careful how you work with logarithms when simplifying logarithmic expressions. Be sure to follow the rules!
Lastly, be sure to study both the algebraic processes, the connection back to exponential functions, evaluating logarithms on the calculator (including using the change of base theorem), and the difference between a common logarithm and a natural logarithm.
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
- 13.2.9 Understand how and when to use the change of base theorem (memorize it!)
- 13.2.10 Know and be able to apply the basic properties of logarithms (memorize them!)
- 13.2.11 Know and be able to apply the product, quotient, and power rules of exponents (memorize them!)
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
Text Notes
Text:
Intro & Inter Algebra for CS 3ed by Blitzer, sect. 12.2-12.3
- ch 12.2 pg 832 The conversion from logarithmic to exponential 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).
- ch 12.2 pg 837+839 MEMORIZE the basic properties of logarithms and how they expand to common logs and natural logs given in the two charts.
- ch 12.3 pg 847-848 MEMORIZE of the rules of logarithms (the product rule, the quotient rule, and the power rule), in both directions "expanding" and "condensing", given in the two charts.
- ch 12.3 pg 847 Take note of the "caution" which warns against some very common errors that can occur when simplifying logarithms.
- ch 12.3 pg 850 MEMORIZE the change of base theorem, it will come in handy over and over. Note that it can be written as logbx = log(x)/log(b) or as logbx = ln(x)/ln(b) with the latter being more common. This theorem is vital for entering some logarithms on your calculator. For instance, log415 can be entered as ln(15)/ln(4) = 1.853 (try it).
- Pg 884-885 has a nice overview of all the properties and rules of logarithms.