Lesson 3

Lesson 3: Understanding Logarithms

Focus

In the early 1600s, John Napier published a way to reduce the tedious process of multiplying large numbers to the easier process of addition. His process involved looking up numbers (which he called logarithms) in a table of values. It took Napier 20 years to create that table of values!

At the time, Napier’s process was very useful to astronomers who needed to multiply large numbers together. Today, however, logarithms are not used for multiplying large numbers; calculators are used instead. That said, there are other uses of logarithms. You will use logarithms in future lessons to solve exponential equations. Before you can solve exponential equations in this new way, you need to explore the concept of logarithms.

In Lesson 1 you studied exponential functions and how they can be used to describe growth or decay. In this lesson you will be introduced to logarithmic functions. These functions can also be used to describe growth or decay. The growth of a tree can be modelled using a logarithmic function. The height of the tree starts with a rapid increase and then the growth rate slows.

Lesson Outcomes

At the end of this lesson you will be able to

• explain the relationship between logarithms and exponents
• express a logarithmic expression as an exponential expression and vice versa
• determine the value of a logarithm

Lesson Question

You will investigate the following question:

• What are logarithms and how is the value of a logarithm determined?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 3 Assignment (Download the Lesson 3 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet

Discover

In the previous lesson you solved exponential equations by guessing and checking or, when possible, ensuring common bases and then equating the exponents. You verified the solution through graphing or substitution. You will use these same skills to understand logarithms and their uses.

Try This 1

1.  
   1. Complete the table provided for the function y = 2x; then graph the function.
      
      

      x	y
      −1
      0
      1
      2
      3
      4

   2. State the domain, range, x-intercept, y-intercept, and equation of asymptotes for the graph.
2.  
   1. Complete the table provided for the inverse function x = 2y; then graph the inverse function.
      
      

      x	y
      	−1
      	0
      	1
      	2
      	3
      	4

   2. State the domain, range, x-intercept, y-intercept, and equation of asymptotes for this inverse relation.
3. Does the inverse graph represent a function? Explain your answer.
4.  
   1. How many 2’s are multiplied together to produce 8? In other words, 2 to what exponent would equal 8?
   2. Looking at the inverse function table or graph, when x = 8 what is the y-value?
   3. What do you notice about your answer to question 4 parts a and b? Why do you think this is?

Save your responses in your course folder.

Share 1

With a partner or group, discuss the following question based on your graph created in Try This 1:

How are the graphs of y = 2x and the inverse, x = 2y, related?

If required, save a record of your discussion in your course folder.

Explore

Your graphs, from Try This 1, of y = 2x and its inverse, x = 2y, may have looked similar to the graphs shown here. The inverse is a function because there is only one x-value for each y-value.

A logarithmic function is the inverse of an exponential function. This means that a logarithmic function is a reflection of the corresponding exponential function in the line y = x.

If you take the inverse function, x = 2y, and let x = 8, the function becomes 8 = 2y. This equation can be described in words as “2 to what exponent equals 8?” Instead of writing it this way, a logarithm could be used to describe the exponent. A logarithm is really the exponent that is required to write a value as a power with a specific base.

When you took the inverse of y = 2x, you produced the function x = 2y. If you were to try to isolate the y-value of x = 2y, you would likely get stuck. Logarithms can help isolate y.

In Lesson 1 you learned that an exponential function is a function in the form y = cx, where c > 0, c ≠ 1. The inverse of the exponential function y = cx would be the logarithmic function in the form y = logc x, where c > 0, c ≠ 1. Logc x is pronounced as “log base c of x”; and log2 x is pronounced as “log base 2 of x.”

In Try This 1 you may have stated the following characteristics about the inverse of the exponential function, the logarithmic function:

• The domain is {x|x > 0, x ∈ R}.
• The range is {y|y ∈ R}.
• The x-intercept is 1.
• There is no y-intercept.
• There is a vertical asymptote at x = 0 (the y-axis).

In the next lesson you will take a closer look at the graphs of logarithmic functions. The rest of this lesson will focus on solving logarithmic expressions.

You have seen how useful the logarithmic form can be to help isolate y in an inverse exponential function. In Try This 2 you will learn more about the logarithmic form and look for patterns between exponential form and logarithmic form.

Try This 2

Complete a table like this one.

Save your responses in your course folder.

Share 2

With a partner or group, discuss the following questions based on Try This 2.

1. Describe the patterns you see in Try This 2.
2. Summarize your discussion by defining a logarithm or by describing the value of a logarithm.

If required, save a record of your discussion in your course folder.

In Try This 2 you may have noticed that the logarithm is equal to the exponent from the exponential form. The relationship between the logarithmic and exponential forms are summarized in the illustration shown.

1

Logarithms with base 10 are often used and are called common logarithms. An example of a common logarithm is when you evaluated log 1000. There is no base written, but it is assumed that the base is 10. So, log 1000 is the same as log10 1000.

View Evaluating Logarithms for more examples of how to evaluate logarithms. Compare this method of evaluation with your table from Try This 2.

Here is another explanation of logarithms: “Introduction to Logarithms.” However, if you feel confident with logarithms, skip the video and move on to Self-Check 1.

Self-Check 1

In Try This 3 you may have found that  could be changed to exponential form and written as follows:

You may have found that logx 49 = 2 could be changed to exponential form and written as follows:

The base of a logarithm is restricted to values greater than zero, so x = −7 is not an acceptable answer. This means x = 7.

Connect

Lesson 3 Assignment

Lesson 3 Summary

In this lesson you explored the concept of logarithms. The inverse of an exponential function, y = cx, c > 0, c ≠ 1, is a logarithmic function, y = logc x, c > 0, c ≠ 1. A logarithm of a number is the exponent by which a given base must be raised to produce that number. You can change between exponential and logarithmic forms.

The logarithmic function is the reflection of the exponential function in the line y = x. The characteristics of the logarithmic function y = logc x, c > 0, c ≠ 1 are as follows:

• The domain is {x|x > 0, x ∈ R}.
• The range is {y|y ∈ R}.
• The x-intercept is 1.
• The vertical asymptote is x = 0, or the y-axis.

In the next lesson you will explore the graphs of different logarithmic functions.