Algebra II End-of-Course Exam Content Standards—Module: Logarithmic Functions
Successful students will be able to define, represent, and model using logarithmic functions. Recognition of the inverse relationship between logarithmic and exponential functions is essential to this concept. They will apply the laws of logarithms, solve logarithmic equations, and use logarithms to solve exponential equations. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and solve rich contextual problems.
L1. Logarithmic expressions and equations
a. Apply the properties of logarithms and use them to manipulate logarithmic expressions.
- Represent logarithmic expressions in exponential form and exponential expressions in logarithmic form.
- Understand that a logarithm is an exponent that depends on the base used.
- The properties of logarithms include those related to powers, products, quotients, and changing the base.
b. Solve logarithmic equations, paying attention to the possibility of extraneous roots.
Example: If log(x - 3) + log(x + 2) = log(14), then
log(x2 - x - 6) = log(14)
x2 - x - 6 = 14
x2 - x - 20 = 0
x = 5 or -4
However, all solutions must be greater than 3 because of domain restrictions on the original logarithmic functions, and the only solution is x = 5.
L2. Logarithmic functions
a. Determine key characteristics of logarithmic functions.
- Key characteristics include domain, range, asymptotes, and end behavior.
b. Represent logarithmic functions using tables, graphs, verbal statements, and equations. Translate among these representations.
c. Describe the effect that changes in the parameters of a logarithmic function have on the shape and position of its graph.
d. Recognize, express, and solve problems that can be modeled using logarithmic functions. Interpret their solutions in terms of the context of the problem.
- Use the properties of logarithms to solve problems involving logarithmic and exponential functions, including problems of growth and decay, chemical pH, and sound or earthquake intensity.
- This includes using and interpreting appropriate units of measurement and precision for the given application.