Chapter 9.4: Logarithmic Functions
Learning Objectives
In this section, you will:
 Convert from logarithmic to exponential form.
 Convert from exponential to logarithmic form.
 Evaluate logarithms.
 Use common logarithms.
 Use natural logarithms.
In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes^{[1]} . One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,^{[2]} like those shown in (Figure). Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale^{[3]} whereas the Japanese earthquake registered a 9.0.^{[4]}
The Richter Scale is a baseten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is times as great! In this lesson, we will investigate the nature of the Richter Scale and the baseten function upon which it depends.
Converting from Logarithmic to Exponential Form
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is where represents the difference in magnitudes on the Richter Scale. How would we solve for
We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve We know that and so it is clear that must be some value between 2 and 3, since is increasing. We can examine a graph, as in (Figure), to better estimate the solution.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in (Figure) passes the horizontal line test. The exponential function is onetoone, so its inverse, is also a function. As is the case with all inverse functions, we simply interchange and and solve for to find the inverse function. To represent as a function of we use a logarithmic function of the form The base logarithm of a number is the exponent by which we must raise to get that number.
We read a logarithmic expression as, “The logarithm with base of is equal to ” or, simplified, “log base of is ” We can also say, “ raised to the power of is ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since we can write We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
Note that the base is always positive.
Because logarithm is a function, it is most correctly written as using parentheses to denote function evaluation, just as we would with However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as Note that many calculators require parentheses around the
We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means and are inverse functions.
Definition of the Logarithmic Function
A logarithm base of a positive number satisfies the following definition.
For
where,
 we read as, “the logarithm with base of ” or the “log base of
 the logarithm is the exponent to which must be raised to get
Also, since the logarithmic and exponential functions switch the and values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,
 the domain of the logarithm function with base
 the range of the logarithm function with base
Can we take the logarithm of a negative number?
No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.
How To
Given an equation in logarithmic form convert it to exponential form.
 Examine the equation and identify
 Rewrite as
Converting from Logarithmic Form to Exponential Form
Write the following logarithmic equations in exponential form.
Show Solution
First, identify the values of Then, write the equation in the form

Here, Therefore, the equation is equivalent to

Here, Therefore, the equation is equivalent to
Try It
Write the following logarithmic equations in exponential form.
Show Solution
 is equivalent to
 is equivalent to
Converting from Exponential to Logarithmic Form
To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base exponent and output Then we write
Converting from Exponential Form to Logarithmic Form
Write the following exponential equations in logarithmic form.
Show Solution
First, identify the values of Then, write the equation in the form

Here, and Therefore, the equation is equivalent to

Here, and Therefore, the equation is equivalent to

Here, and Therefore, the equation is equivalent to
Try It
Write the following exponential equations in logarithmic form.
Show Solution
 is equivalent to
 is equivalent to
 is equivalent to
Evaluating Logarithms
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider We ask, “To what exponent must be raised in order to get 8?” Because we already know it follows that
Now consider solving and mentally.
 We ask, “To what exponent must 7 be raised in order to get 49?” We know Therefore,
 We ask, “To what exponent must 3 be raised in order to get 27?” We know Therefore,
Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate mentally.
 We ask, “To what exponent must be raised in order to get ” We know and so Therefore,
How To
Given a logarithm of the form evaluate it mentally.
 Rewrite the argument as a power of
 Use previous knowledge of powers of identify by asking, “To what exponent should be raised in order to get ”
Solving Logarithms Mentally
Solve without using a calculator.
Show Solution
First we rewrite the logarithm in exponential form: Next, we ask, “To what exponent must 4 be raised in order to get 64?”
We know
Therefore,
Try It
Solve without using a calculator.
Show Solution
(recalling that )
Evaluating the Logarithm of a Reciprocal
Evaluate without using a calculator.
Show Solution
First we rewrite the logarithm in exponential form: Next, we ask, “To what exponent must 3 be raised in order to get ”
We know but what must we do to get the reciprocal, Recall from working with exponents that We use this information to write
Therefore,
Try It
Evaluate without using a calculator.