4.2: Properties of Logarithms

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Objectives:

• Understand and practice using common log and natural log
• Rewrite and evaluate statements with logs using exponents
• Rewrite and evaluate statements with exponents using logs.
• Discover and utilize properties of logarithms

Important Items

Definitions:

common log, natural log

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Understand basic properties of logarithms and how to evaluate them

Introduce logs as the inverse functions of exponentials. Then use an example like the following to talk about how to find the inverse of exponential functions.

- Example: Suppose $P=Q(t)=100(2)^t$.

• If $Q(0)=100$, then what is $Q^{-1}(100)$?
• Solve $Q^{-1}(P)=3$ by using the given equation for $Q(t)$.

We can continue to use $Q(t)$ to evaluate and solve for $Q^{-1}(P)$, but would be nice if we could write an equation for $Q^{-1}(P)$. In fact, we can do that as long as we can find a way to "undo" $Q(t)$ by working backwards.

Students tend to think that roots are the inverse of exponentials. Make sure to address and emphasize the difference between the two.

The purpose of today's lesson is to learn how to "undo" this final step.

logarithmic function is the inverse function of the special exponential 
function, $Q(t)=b^t$. In other words, if $P=b^t$, then $\log_b(P)=t$.
 

Do several examples of how to evaluate logs. Also graph several of these examples so that students gain an understanding of the graphic qualities of a logarithmic function and how it relates to its inverse exponential function.

• Example:

• Example:

Have students do Problems 1-3.

Throughout class, pause and have students present their solutions. Try to ensure that all students feel comfortable with the basic concept of a logarithm being an inverse of an exponential.

Understand and practice using common log and natural log

Explain that base 10 and base $e$ are so common that we developed specific mathematical notation to denote when the base is either 10 or $e$.

* $\log(P)$ is the power of 10 that gives $P$. So if $t=\log(P)$, then $10^t=P$.
* $\ln(P)$ is the power of $e$ that gives $P$. So if $t=\ln(P)$, then $e^t=P$.

Some students forget about the base matters, since we "drop" it when we use the common and natural log notations. Emphasize that other bases can be used, but $\log(P)=\log_{10}(P)$ and $\ln(P)=\log_e(P)$.

Do an example similar to Problem 4.

-Example:

• Have students do Problems 4 and 5.

Rewrite and evaluate statements with logs using exponents.

• Do an example similar to Problem 6.

-Example:

• Have students do Problem 6.

Rewrite and evaluate statements with exponents using logs.

• Do some examples similar to Problems 7 and 8.

-Example:

Example:

• Have students do Problems 7 and 8

Discover and utilize properties of logarithms.

Have students do Problem 9.

Use Problem 9 to talk about the following properties of logarithms.

Properties of Logarithms:
* $\log_b(xy)=\log_b(x)+\log_b(y)$
* $\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)$
* $\log_b(x^y)=y\log_b(x)$
* $\log_b(b^x)=x$
* $b^{\log_b(x)}=x$

To explain why the first three properties are true, relate them to the corresponding properties of exponentials. To explain why the last two properties are true, talk about the connection between inverse functions (it may be helpful to draw bubble diagrams).