Section 4.3Logarithmic Functions

Corequisite Skills review (optional warm-up)

Learning Objectives

• Convert between exponential and logarithmic form. (IA 10.3.1)
• Evaluate logarithmic functions. (IA 10.3.2)

Objective 1: Convert between exponential and logarithmic form. (IA 10.3.1)

Practice Makes Perfect

P1

Graph the exponential function \(f(x)={2}^{x}\) by making a table.

Table (fill in)

\(x\)
\(y=f(x)\)

• ⓐ Is it one-to-one?
• ⓑ Domain?
• ⓒ Range?
• ⓓ Graph the inverse of \(f(x)={2}^{x}\) on the grid above by interchanging x and y coordinates in the table.
• ⓔ Is the inverse a one-to-one function?
• ⓕ Domain?
• ⓖ Range?

Warm-up Example 1

Find the inverse of \(f(x)={2}^{x}\)

Show solution

• Rewrite with \(y=f(x)\): \(y={2}^{x}\)
• Interchange the variables \(x\) and \(y\): \(x={2}^{y}\)
• Solve for \(y\): Oops! We have no way to solve for \(y\).

We give y a new notation:

\[y=\log_2 x\]

"\(y=\log_2 x\)" read "the logarithm, base 2, of x," means "the power to which we raise 2 to get x."

The function \(y=\log_a x\) is equivalent to \(a^y=x\) is the logarithmic function with base \(a\), where \(a>0\), \(x>0\).

Since the equations \(y=\log_a x\) and \(x=a^y\) are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth, let's take a close look at the equations. Notice the positions of the exponent and base.

If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, "base to the exponent gives us the number."

Warm-up Example 2

Convert between exponential and logarithmic form.

ⓐ Convert to logarithmic form: \(2^3=8\)

Show solution

Identify the base and the exponent: the base is 2 and the exponent is 3.

Then we have \(3=\log_2 8\).

ⓑ Convert to exponential form: \(\log_b m=a\)

Show solution

Identify the base and the exponent: the base is b and the exponent is a.

Then we have \(b^a=m\).

Practice Makes Perfect

Convert between exponential and logarithmic form.

Remember these logarithmic notations to help complete the following:
Common Logarithm \(\log x=\log_{10} x\)
Natural Logarithm \(\ln x=\log_e x\)

P2

Convert to logarithmic form.

• ⓐ \(8=2^x\)
• ⓑ \(10^{-2}=0.01\)
• ⓒ \(e^x=40\)

P3

Convert to exponential form.

• ⓐ \(\log_{81} 3=4\)
• ⓑ \(\ln 1=0\)
• ⓒ \(\log 10000=4\)

Objective 2: Evaluate logarithmic functions (IA 10.3.2).

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential form.

Warm-up Example 3

Find the value of x: ⓐ \(\log_x 36=2\), ⓑ \(\log_4 x=3\), and ⓒ \(\log_{1/2} \frac{1}{8}=x.\)

Show solution

ⓐ
\(\log_x 36=2\)
Convert to exponential form: \(x^2=36\)
Solve the quadratic: \(x=6,\ \cancel{x=-6}\)
The base of a logarithmic function must be positive, so we eliminate \(x=-6\). \(x=6\). Therefore, \(\log_6 36=2\).

ⓑ
\(\log_4 x=3\)
Convert to exponential form: \(4^3=x\)
Simplify: \(x=64\). Therefore, \(\log_4 64=3\).

ⓒ
\(\log_{1/2} \frac{1}{8}=x\)
Convert to exponential form: \(\left(\frac{1}{2}\right)^x=\frac{1}{8}\)
Rewrite \(\frac{1}{8}\) as \(\left(\frac{1}{2}\right)^3\): \(\left(\frac{1}{2}\right)^x=\left(\frac{1}{2}\right)^3\)
With the same base, the exponents must be equal: \(x=3\). Therefore, \(\log_{1/2} \frac{1}{8}=3\).

Practice Makes Perfect

P4

Find the value of x.

• ⓐ \(\log_{25} x=2\)
• ⓑ \(\log_x 4=2\)
• ⓒ \(\log_x \frac{1}{3}=2\)

P5

Evaluate each of the following.

• ⓐ \(\log_{100} 10\)
• ⓑ \(\log 0.1\)
• ⓒ \(\log_2 4\)
• ⓓ \(\log_1 20\)
• ⓔ \(\log_4 4\)
• ⓕ \(\log_{1/9} 3\)
• ⓖ \(\log_{\sqrt2} 2\)
• ⓗ \(\ln e^{-5}\)

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings, like those shown in Figure 1. 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 whereas the Japanese earthquake registered a 9.0.

The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is

\[10^{8-4}=10^4=10{,}000\]

times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten 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 \(10^x=500\), where \(x\) represents the difference in magnitudes on the Richter Scale. How would we solve for \(x\)?

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve \(10^x=500\). We know that \(10^2=100\) and \(10^3=1000\), so it is clear that \(x\) must be some value between 2 and 3, since \(y=10^x\) is increasing. We can examine a graph, as in Figure 2, 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 2 passes the horizontal line test. The exponential function \(y=b^x\) is one-to-one, so its inverse, \(x=b^y\) is also a function. As is the case with all inverse functions, we simply interchange \(x\) and \(y\) and solve for \(y\) to find the inverse function. To represent \(y\) as a function of \(x\), we use a logarithmic function of the form \(y=\log_b(x)\). The base \(b\) logarithm of a number is the exponent by which we must raise \(b\) to get that number.

We read a logarithmic expression as, "The logarithm with base \(b\) of \(x\) is equal to \(y\)," or, simplified, "log base \(b\) of \(x\) is \(y\)." We can also say, "\(b\) raised to the power of \(y\) is \(x\)," 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 \(2^5=32\), we can write \(\log_2 32=5\). 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:

\[\log_b(x)=y \Leftrightarrow b^y=x,\quad b>0,\ b\ne1\]

Note that the base \(b\) is always positive.

Because logarithm is a function, it is most correctly written as \(\log_b(x)\), using parentheses to denote function evaluation, just as we would with \(f(x)\). 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 \(\log_b x\). Note that many calculators require parentheses around the \(x\).

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 \(y=\log_b(x)\) and \(y=b^x\) are inverse functions.

Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base \(b\), exponent \(x\), and output \(y\). Then we write \(x=\log_b(y)\).

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider \(\log_2 8\). We ask, "To what exponent must 2 be raised in order to get 8?" Because we already know \(2^3=8\), it follows that \(\log_2 8=3\).

Now consider solving \(\log_7 49\) and \(\log_3 27\) mentally.

• We ask, "To what exponent must 7 be raised in order to get 49?" We know \(7^2=49\). Therefore, \(\log_7 49=2\)
• We ask, "To what exponent must 3 be raised in order to get 27?" We know \(3^3=27\). Therefore, \(\log_3 27=3\)

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let's evaluate \(\log_{2/3}\frac{4}{9}\) mentally.

• We ask, "To what exponent must \(\frac{2}{3}\) be raised in order to get \(\frac{4}{9}\)?" We know \(2^2=4\) and \(3^2=9\), so \(\left(\frac{2}{3}\right)^2=\frac{4}{9}\). Therefore, \(\log_{2/3}\left(\frac{4}{9}\right)=2\).

Using Common Logarithms

Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means \(\log_2(x)\). However, in mathematics it almost always means the common logarithm of 10. In other words, the expression \(\log(x)\) often means \(\log_{10}(x)\).

Using Natural Logarithms

The most frequently used base for logarithms is \(e\), the value of which is approximately 2.71828. Base \(e\) logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base \(e\) logarithm, \(\log_e(x)\), has its own notation, \(\ln(x)\).

Most values of \(\ln(x)\) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, \(\ln1=0\). For other natural logarithms, we can use the ln key that can be found on most scientific calculators. We can also find the natural logarithm of any power of \(e\) using the inverse property of logarithms.

Key Equations

Key Concepts

• The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
• Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See Example 1.
• Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm. See Example 2.
• Logarithmic functions with base \(b\) can be evaluated mentally using previous knowledge of powers of \(b\). See Example 3 and Example 4.
• Common logarithms can be evaluated mentally using previous knowledge of powers of 10. See Example 5.
• When common logarithms cannot be evaluated mentally, a calculator can be used. See Example 6.
• Real-world exponential problems with base 10 can be rewritten as a common logarithm and then evaluated using a calculator. See Example 7.
• Natural logarithms can be evaluated using a calculator. See Example 8.