Interlude: Introduction to Logarithms
Contents
Objectives
- Understand basic properties of logarithms and how to evaluate them
Important Items
Definitions:
logarithmic function
Notes:
This lesson is considered to be the necessary background to \S4.2. Logarithms are difficult for many students! Furthermore, while the remainder of Chapter 4 focuses on only common logs and natural logs, we will discuss logs with any base in this lesson.
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.
Comments
- Make sure students are aware that from 4.2 on, they will be expected to only use common log and natural log. The purpose of the interlude was to introduce logs in a general sense so that they don't think $10^x$ and $e^x$ are the only exponential functions with inverses, but logs with different bases won't be used for the rest of the semester.
- In my class, although inverses had come up regularly in the past few weeks, students are still not completely comfortable with them. Writing down in a box to the side the fact that $Q(t) = P$ and $Q^{-1}(P) = t$, and showing how this was used in defining the logarithm via color coding really seemed to get the point across that the logarithm is the inverse of the exponential and also helped highlight the relationship between the two (specifically using the exponential function to solve logarithmic expressions). -Elizabeth C
