Exponential and Logarithmic Functions¶

Goals for Section 3.1 of the book¶

Students should be able to do each of the following:

Determine if a given function is one-to-one. The function can be given in graphical, tabular, or algebraic forms.
Explicitly show that a function is one-to-one or demonstrate why a function is not one-to-one.
Use the horizontal line test to determine if a function is one-to-one.
Determine the inverse of a given function. The function can be given in tabular or algebraic forms.
Determine the domain and range of the inverse of a function. The function can be given in graphical, tabular, or algebraic forms.
Limit the domain of a function that is not one-to-one so that an
inverse can be defined on the resulting restricted domain.

Students should be able to do each of the following:

Use the definition of an exponential function to define a function describing a situation given in written, verbal, or graphical form.
Use the properties and operations of exponential functions to solve for any variable in an expression that has exponential functions.
Be able to graph exponential functions. Be able to compare and identify different exponential functions whose parameters differ.
Convert and write any exponential function using base e.
Solve compound interest problems given a written description of the situation.
Solve exponential growth/decay problems given a written description of the situation. Be able to identify if a situation results in either growth or decay.
Given a written description determine if an exponential function or a logistic growth function is appropriate.

Students should be able to do each of the following:

Use the logarithm function to solve for a variable in an equation that has exponential terms.
Determine the domain and range of a simple function that contains logarithmic terms.
Determine the inverse of a simple exponential function.
Determine the inverse of a function that contains logarithmic terms.
Graph basic logarithmic functions.
Solve for a variable in an equation that contains logarithmic terms.
Recognize that \(\log(x)=\log_{10}(x)\).
Recognize that \(\ln(x)=\log_e(x)\).

Students should be able to do each of the following:

Use the properties of logarithms to solve for a variable in a
variety of more complex forms:

\(\log_a(u\cdot v) = \log_a(u) + \log_a(v),\)

\(\log_a\left(\frac{u}{v}\right) = \log_a(u) - \log_a(v),\)

\(\log_a\left(u^r\right) = r\log_a(u).\)
Solve for a variable when multiple logarithms with different bases are present in an expression. (Either using the substitution method or the change of base formula.)
Change the base for a logarithm to either base e or base 10 so an approximation can be found using a calculator.

Students should be able to do each of the following:

Solve equations with multiple exponential terms.
Determine inverses of complicated functions that have either exponential or logarithmic terms.
Manipulate equations with exponentials and transform them into other forms. For example, transform an expression with exponential terms into a quadratic equation.
Define relationships given written descriptions that include wither exponential or logarithmic terms.
Determine the value of any or all parameters in a compound interest
problem given a written description of the situation.
Determine the value of any or all parameters in an exponential
growth/decay problem given a written description of the situation. Be able to identify if a situation results in either growth or decay.
Determine the value of any or all parameters given a written
description of a logistic growth function.

Students should be able to do each of the following: