4.4: Applications of the Logarithm

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Objectives

• Identify the domain and range of $t=a\log(P-b)$ and $t=c\ln(P-d)$ and understand their relationship to the domain and range of $P=10^{t/a}+b$ and $P=e^{t/c}+d$.
• Manipulate real-world examples that can be modeled by a logarithm

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Lesson Guide

Kelsey's Alternative Approach

Whether you use Kelsey's approach or this approach, consider at some point using the following demo to show how the log base changes the graph (this app also exists in the book in 8.2, which may actually be better to use):

Warm-Up

Have students do Problems 1 and 2.

Identify the domain and range of $t=a\log(P-b)$ and $t=c\ln(P-d)$ and understand their relationship to the domain and range of $P=10^{t/a}+b$ and $P=e^{t/c}+d$.

Graph several logarithmic functions of the form $t=a\log(P-b)$ and $t=c\ln(P-d).$ Ask students to comment on the common characteristics of these graphs. Ask them to find the exponential function for which they are the inverse. Compare the graphs of the exponential with the graph of its inverse to help students gain intuition on their relationship.

-Example:

-Example:

Emphasize that the domain and range of a function are switched for that function's inverse. Show this carefully for an exponential and its inverse function.

Discuss asymptotes, and remind students of the fact that an exponential of the form $P=(b)^t$ has a horizontal asymptote at $P=0.$ So a logarithmic function of the form $t=\log_b(P)$ has a vertical asymptote at $P=0.$ Generalize this to asymptotes of other logarithmic functions like $a\log(P-b)$.

• Note: There are no problems in this section asking students about graphs or asymptotes. Consider asking students to graph one or more of the functions they find in these problems, and ask them to interpret the meaning of the asymptote in the context of the problem.

Have students do Problem 3.

Manipulate real-world examples that can be modeled by a logarithm

While we don't formally introduce the idea of orders of magnitude, students still struggle with the idea of how to compare two quantities. One way you can help students understand this is by doing an example that just focussing on calculating how many times more one quantity is than another.

-Example: Suppose that Susan has saved \$10,000 and Marc has saved \$6,300. How many times more money has Susan saved than Marc? \begin{align*} 6300x&=10000 \\ x&=\frac{10000}{6300} \\ x&\approx1.5873 \end{align*} What if all we knew was that Susan has saved $D_S$ dollars and Marc has saved $D_M$ dollars? How would we compute how many times more money Susan saved than Marc? \begin{align*} D_Mx&=D_S \\ x&=\frac{D_S}{D_M} \end{align*}

Have students do Problems 4 and 5. Students may not realize that they will need the formula given to them in Problem 4 in order to do Problem 5.

Since students often struggle with using $A_0$ in Problem 6, let students know that $A_0$ is \textbf{not} a variable, but an unknown, constant quantity.

Have students do Problem 6.