2.4 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.
  • Use logarithmic properties.
Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag.

Figure 1  Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)

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 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 [3] whereas the Japanese earthquake registered a 9.0[4]

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 [latex]{10}^{8-4}={10}^{4}=10,000[/latex] times as great! In this section, 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 [latex]{10}^{x}=500,[/latex] where [latex]x[/latex] represents the difference in magnitudes on the Richter Scale. How would we solve for [latex]x?[/latex]

We have only learned a graphical method for approximating solutions of exponential equations. None of the algebraic tools discussed so far is sufficient to solve [latex]{10}^{x}=500.[/latex] We know that [latex]{10}^{2}=100[/latex] and [latex]{10}^{3}=1000,[/latex] so it is clear that [latex]x[/latex] must be some value between 2 and 3, since [latex]y={10}^{x}[/latex] is increasing. We can examine a graph, as in Figure 2, to better estimate the solution.

Graph of the intersections of the equations y=10^x and y=500.

Figure 2

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 [latex]y={b}^{x}[/latex] is one-to-one, so its inverse is also a function. We use a logarithmic function of the form [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] to describe the inverse.  The base [latex]b[/latex] logarithm of a number is the exponent by which we must raise [latex]b[/latex] to get that number.

We read a logarithmic expression as, “The logarithm with base [latex]b[/latex] of [latex]x[/latex] is equal to [latex]y,[/latex]” or, simplified, “log base [latex]b[/latex] of [latex]x[/latex] is [latex]y.[/latex]” We can also say, “[latex]b[/latex] raised to the power of [latex]y[/latex] is [latex]x,[/latex]” 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 [latex]{2}^{5}=32,[/latex] we can write [latex]{\mathrm{log}}_{2}32=5.[/latex] 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:

[latex]{\mathrm{log}}_{b}\left(x\right)=y⇔{b}^{y}=x,\text{ }b>0,b\ne 1[/latex][latex]\\[/latex]

Note that the base [latex]b[/latex] is always positive.

Because logarithm is a function, it is most correctly written as [latex]{\mathrm{log}}_{b}\left(x\right),[/latex] using parentheses to denote function evaluation, just as we would with [latex]f\left(x\right).[/latex] 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 [latex]{\mathrm{log}}_{b}x.[/latex] Note that many calculators require parentheses around the [latex]x.[/latex]

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.

Definition

A logarithm base [latex]b[/latex] of a positive number [latex]x[/latex] satisfies the following definition.

For [latex]x>0,\text{ }b>0,\text{ }b\ne 1,[/latex]  [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is equivalent to [latex]{b}^{y}=x.[/latex]

  • We read [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] as, “the logarithm with base [latex]b[/latex] of [latex]x[/latex]” or the “log base [latex]b[/latex] of [latex]x."[/latex]
  • The logarithm [latex]y[/latex] is the exponent to which [latex]b[/latex] must be raised to get [latex]x.[/latex]

Also, since the logarithmic and exponential functions are inverses of each other, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base [latex]b[/latex] is [latex]\left(0,\infty \right),[/latex] and
  • the range of the logarithm function with base [latex]b[/latex] is [latex]\left(-\infty ,\infty \right).[/latex]

 

Q&A

Can we take the logarithm of a negative number?

No. We are working with functions of real numbers. Because the base of an exponential function is always a positive real number, no power of that base can ever be a negative real number. We can never take the logarithm of a negative real 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 [latex]{\mathrm{log}}_{b}\left(x\right)=y,[/latex] convert it to exponential form.

  1. Examine the equation [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] and identify [latex]b,\text{ }y,[/latex] and [latex]x.[/latex]
  2. Rewrite [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex] as [latex]{b}^{y}=x.[/latex]

Example 1:  Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.

  1. [latex]{\mathrm{log}}_{6}\left(\sqrt[\leftroot{1}\uproot{2} ]{6}\right)=\frac{1}{2}[/latex]
  2. [latex]{\mathrm{log}}_{10}\left(100\right)=2[/latex]

Try it #1

Write the following logarithmic equations in exponential form.

  1. [latex]{\mathrm{log}}_{10}\left(1,000,000\right)=6[/latex]
  2. [latex]{\mathrm{log}}_{5}\left(25\right)=2[/latex]

Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base [latex]b,[/latex] exponent [latex]x,[/latex] and output [latex]y[/latex] in [latex]b^x=y.[/latex] Then we write  [latex]x={\mathrm{log}}_{b}\left(y\right).[/latex]

Example 2:  Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

  1. [latex]{2}^{3}=8[/latex]
  2. [latex]{10}^{4}=10000[/latex]
  3. [latex]{10}^{-4}=\frac{1}{10,000}[/latex]

Try it #2

Write the following exponential equations in logarithmic form.

  1. [latex]{1.1}^{2}=1.21[/latex]
  2. [latex]{5}^{3}=125[/latex]
  3. [latex]{10}^{-1}=\frac{1}{10}[/latex]

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\mathrm{log}}_{2}\left(8\right).[/latex] We ask, “To what exponent must [latex]2[/latex] be raised in order to get 8?” Because we already know [latex]{2}^{3}=8,[/latex] it follows that [latex]{\mathrm{log}}_{2}\left(8\right)=3.[/latex]

Now consider solving [latex]{\mathrm{log}}_{7}\left(49\right)[/latex] and [latex]{\mathrm{log}}_{3}\left(27\right)[/latex] mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know [latex]{7}^{2}=49.[/latex] Therefore, [latex]{\mathrm{log}}_{7}\left(49\right)=2.[/latex]
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know [latex]{3}^{3}=27.[/latex]Therefore, [latex]{\mathrm{log}}_{3}\left(27\right)=3.[/latex]

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate [latex]{\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)[/latex] mentally.

  • We ask, “To what exponent must [latex]\frac{2}{3}[/latex] be raised in order to get [latex]\left(\frac{4}{9}\right)?[/latex]”  We know [latex]{2}^{2}=4[/latex] and [latex]{3}^{2}=9,[/latex] so [latex]{\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}.[/latex] Therefore, [latex]{\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2.[/latex]

[latex]\\[/latex]

How To

Given a logarithm of the form [latex]y={\mathrm{log}}_{b}\left(x\right),[/latex] evaluate it mentally.

  1. Rewrite the argument [latex]x[/latex] as a power of [latex]b:[/latex]  [latex]{b}^{y}=x.[/latex]
  2. Use previous knowledge of powers of [latex]b[/latex] identify [latex]y[/latex] by asking, “To what exponent should [latex]b[/latex] be raised in order to get [latex]x?[/latex]”

Example 3:  Solving Logarithms Mentally

Solve [latex]y={\mathrm{log}}_{4}\left(64\right)[/latex] without using a calculator.

Try it #3

Solve [latex]y={\mathrm{log}}_{121}\left(11\right)[/latex] without using a calculator.

Example 4:  Evaluating the Logarithm of a Reciprocal

Evaluate [latex]y={\mathrm{log}}_{10}\left(\frac{1}{100}\right)[/latex] without using a calculator.

Try it #4

Evaluate [latex]y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)[/latex] without using a calculator.

Using Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression [latex]\mathrm{log}\left(x\right)[/latex] means [latex]{\mathrm{log}}_{10}\left(x\right).[/latex] We call a base-10 logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Definition

A common logarithm is a logarithm with base [latex]10.[/latex] We write [latex]{\mathrm{log}}_{10}\left(x\right)[/latex] simply as [latex]\mathrm{log}\left(x\right).[/latex] The common logarithm of a positive number [latex]x[/latex] satisfies the following definition.

For [latex]x>0,[/latex] [latex]y=\mathrm{log}\left(x\right)[/latex] is equivalent to [latex]{10}^{y}=x.[/latex]

  • We read [latex]\mathrm{log}\left(x\right)[/latex] as, “the logarithm with base [latex]10[/latex] of [latex]x[/latex]” or “log base 10 of [latex]x.[/latex]”
  • The logarithm [latex]y[/latex] is the exponent to which [latex]10[/latex] must be raised to get [latex]x.[/latex]

How To

Given a common logarithm of the form [latex]y=\mathrm{log}\left(x\right),[/latex] evaluate it mentally.

  1. Rewrite the argument [latex]x[/latex] as a power of [latex]10:[/latex]  [latex]{10}^{y}=x.[/latex]
  2. Use previous knowledge of powers of [latex]10[/latex] to identify [latex]y[/latex] by asking, “To what exponent must [latex]10[/latex] be raised in order to get [latex]x?[/latex]”

Example 5:  Finding the Value of a Common Logarithm Mentally

Evaluate [latex]y=\mathrm{log}\left(1000\right)[/latex] without using a calculator.

Try it #5

Evaluate [latex]y=\mathrm{log}\left(1,000,000\right).[/latex]

How To

Given a common logarithm with the form [latex]y=\mathrm{log}\left(x\right),[/latex] evaluate it using a calculator.

  1. Press [LOG].
  2. Enter the value given for [latex]x,[/latex] followed by [ ) ].
  3. Press [ENTER].

Example 6:  Finding the Value of a Common Logarithm Using a Calculator

Evaluate [latex]y=\mathrm{log}\left(321\right)[/latex] to four decimal places using a calculator.

Try it #6

Evaluate [latex]y=\mathrm{log}\left(123\right)[/latex] to four decimal places using a calculator.

Example 7:  Rewriting and Solving a Real-World Exponential Model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation [latex]{10}^{x}=500[/latex] represents this situation, where [latex]x[/latex] is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Try it #7

The amount of energy released from one earthquake was [latex]\text{8,500}[/latex] times greater than the amount of energy released from another. The equation [latex]{10}^{x}=8500[/latex] represents this situation, where [latex]x[/latex] is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Using Natural Logarithms

The most frequently used base for logarithms is [latex]e.[/latex] Base [latex]e[/latex] logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base [latex]e[/latex] logarithm, [latex]{\mathrm{log}}_{e}\left(x\right),[/latex] has its own notation, [latex]\mathrm{ln}\left(x\right).[/latex]

Most values of [latex]\mathrm{ln}\left(x\right)[/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex]\mathrm{ln}\left(1\right)=0.[/latex] 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 [latex]e[/latex] using the inverse property of logarithms.

Definition

A natural logarithm is a logarithm with base [latex]e.[/latex] We write [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] simply as [latex]\mathrm{ln}\left(x\right).[/latex] The natural logarithm of a positive number [latex]x[/latex] satisfies the following definition.

For [latex]x>0,[/latex] [latex]y=\mathrm{ln}\left(x\right)[/latex] is equivalent to [latex]{e}^{y}=x.[/latex]

  • We read [latex]\mathrm{ln}\left(x\right)[/latex] as, “the logarithm with base [latex]e[/latex] of [latex]x[/latex]” or “the natural logarithm of [latex]x.[/latex]”
  • The logarithm [latex]y[/latex] is the exponent to which [latex]e[/latex] must be raised to get [latex]x.[/latex]

How To

Given a natural logarithm with the form [latex]y=\mathrm{ln}\left(x\right),[/latex] evaluate it using a calculator.

  1. Press [LN].
  2. Enter the value given for [latex]x,[/latex] followed by [ ) ].
  3. Press [ENTER].

Example 8:  Evaluating a Natural Logarithm Using a Calculator

Evaluate [latex]y=\mathrm{ln}\left(500\right)[/latex] to four decimal places using a calculator.

Try it #8

Evaluate:  a. [latex]\mathrm{ln}\left(-500\right).[/latex]  b. [latex]\mathrm{ln}\left(8\right).[/latex]

Properties of Logarithms

In applications, equation solving and advance mathematics, it is often easier to work with simplified logarithmic expressions.  We will next study the properties of logarithms which allow logarithmic expressions to be written in multiple ways so that they can be interpreted from multiple view points.

The logarithmic and exponential functions with the same base “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.

[latex]\begin{array}{l}{\mathrm{log}}_{b}1=0\\ {\mathrm{log}}_{b}b=1\\\end{array}[/latex]

For example, [latex]{\mathrm{log}}_{5}1=0[/latex] since [latex]{5}^{0}=1,[/latex] and [latex]{\mathrm{log}}_{5}5=1[/latex] since [latex]{5}^{1}=5.[/latex]

Next, we have the inverse property.

[latex]\begin{align*}{\mathrm{log}}_{b}\left({b}^{x}\right)&=x\hfill \\{b}^{{\mathrm{log}}_{b}x}&=x,\text{ }x>0\hfill \end{align*}[/latex]

Since the functions [latex]y=e{}^{x}[/latex] and [latex]y=\mathrm{ln}\left(x\right)[/latex] are inverse functions, we know that the composition of the functions produce the identity over the appropriate domain.  Therefore, [latex]\mathrm{ln}\left({e}^{x}\right)=x[/latex] for all [latex]x[/latex] and [latex]e{}^{\mathrm{ln}\left(x\right)}=x[/latex] for [latex]x>0.[/latex] Similarly, [latex]\mathrm{log}\left({10}^{x}\right)=x[/latex] for all [latex]x[/latex] and [latex]10{}^{\mathrm{log}\left(x\right)}=x[/latex] for [latex]x>0.[/latex] For example, to evaluate [latex]\mathrm{log}\left(100\right),[/latex] we can rewrite the logarithm as [latex]{\mathrm{log}}_{10}\left({10}^{2}\right),[/latex] and then apply the inverse property [latex]{\mathrm{log}}_{b}\left({b}^{x}\right)=x[/latex] to get [latex]{\mathrm{log}}_{10}\left({10}^{2}\right)=2.[/latex] To evaluate [latex]{e}^{\mathrm{ln}\left(7\right)},[/latex] we can rewrite the logarithm as [latex]{e}^{{\mathrm{log}}_{e}7},[/latex] and then apply the inverse property [latex]{b}^{{\mathrm{log}}_{b}x}=x[/latex] to get [latex]{e}^{{\mathrm{log}}_{e}7}=7.[/latex]

Product Rule of Logarithms

Recall that we use the product rule of exponents to combine the product of exponents by adding: [latex]{x}^{a}{x}^{b}={x}^{a+b}.[/latex] We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.

Given any real number [latex]x[/latex] and positive real numbers [latex]M,\text{ }N,[/latex] and [latex]b,[/latex] where [latex]b\ne 1,[/latex] we will show

[latex]{\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right).[/latex][latex]\\[/latex]

Let [latex]m={\mathrm{log}}_{b}\left(M\right)[/latex] and [latex]n={\mathrm{log}}_{b}\left(N\right).[/latex] In exponential form, these equations are [latex]{b}^{m}=M[/latex] and [latex]{b}^{n}=N.[/latex] It follows that

[latex]\begin{array}{lll}{\mathrm{log}}_{b}\left(MN\right)\hfill & ={\mathrm{log}}_{b}\left({b}^{m}{b}^{n}\right)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\hfill & \text{Substitute for }M\text{ and }N.\hfill \\ \hfill & ={\mathrm{log}}_{b}\left({b}^{m+n}\right)\hfill & \text{Apply the product rule for exponents}.\hfill \\ \hfill & =m+n\hfill & \text{Apply the inverse property of logs}.\hfill \\ \hfill & ={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)\hfill & \text{Substitute for }m\text{ and }n.\hfill \end{array}[/latex]

 

Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider [latex]{\mathrm{log}}_{b}\left(wxyz\right).[/latex] Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

[latex]{\mathrm{log}}_{b}\left(wxyz\right)={\mathrm{log}}_{b}\left(w\right)+{\mathrm{log}}_{b}\left(x\right)+{\mathrm{log}}_{b}\left(y\right)+{\mathrm{log}}_{b}\left(z\right)[/latex]
[latex]\\[/latex]

How To

Given the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.

  1. Factor the argument completely, expressing each whole number factor as a product of prime numbers.
  2. Write the equivalent expression by summing the logarithms of each factor.

Example 9:  Using the Product Rule for Logarithms

Expand [latex]{\mathrm{log}}_{3}\left(30x\left(3x+4\right)\right).[/latex]

Try it #9

Expand [latex]{\mathrm{log}}_{b}\left(10k\right).[/latex]

Quotient Rule for Logarithms

For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: [latex]\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}.[/latex] The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.

Given any real number [latex]x[/latex] and positive real numbers [latex]M,[/latex] [latex]N,[/latex] and  [latex]b,[/latex] where [latex]b\ne 1,[/latex] we will show

[latex]{\mathrm{log}}_{b}\left(\frac{M}{N}\right)\text{=}{\mathrm{log}}_{b}\left(M\right)-{\mathrm{log}}_{b}\left(N\right).[/latex][latex]\\[/latex]

Let [latex]m={\mathrm{log}}_{b}\left(M\right)[/latex] and [latex]n={\mathrm{log}}_{b}\left(N\right).[/latex] In exponential form, these equations are [latex]{b}^{m}=M[/latex] and [latex]{b}^{n}=N.[/latex] It follows that

[latex]\begin{array}{lll}{\mathrm{log}}_{b}\left(\frac{M}{N}\right)\hfill & ={\mathrm{log}}_{b}\left(\frac{{b}^{m}}{{b}^{n}}\right)\text{ }\text{ }\text{ }\text{ }\text{ }\hfill & \text{Substitute for }M\text{ and }N.\hfill \\ \hfill & ={\mathrm{log}}_{b}\left({b}^{m-n}\right)\hfill & \text{Apply the quotient rule for exponents}.\hfill \\ \hfill & =m-n\hfill & \text{Apply the inverse property of logs}.\hfill \\ \hfill & ={\mathrm{log}}_{b}\left(M\right)-{\mathrm{log}}_{b}\left(N\right)\hfill & \text{Substitute for }m\text{ and }n.\hfill \end{array}[/latex][latex]\\[/latex]

For example, to expand [latex]\mathrm{log}\left(\frac{2{x}^{2}+6x}{3x+9}\right),[/latex] we must first express the quotient in lowest terms. Factoring and canceling we get,

[latex]\begin{align*}\mathrm{log}\left(\frac{2{x}^{2}+6x}{3x+9}\right)&=\mathrm{log}\left(\frac{2x\left(x+3\right)}{3\left(x+3\right)}\right)\hfill && \textrm{Factor}.\hfill \\ \text{ }&=\mathrm{log}\left(\frac{2x}{3}\right)\hfill && \textrm{Cancel the common factors}.\hfill \end{align*}[/latex][latex]\\[/latex]
When the common factors are canceled, keep in mind that [latex]x=-3[/latex] is not in the domain of the original function and should be noted in the work as the expression is simplified.

Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule to the first term.

[latex]\begin{align*}\mathrm{log}\left(\frac{2x}{3}\right)&=\mathrm{log}\left(2x\right)-\mathrm{log}\left(3\right)\hfill \\ \text{ }&=\mathrm{log}\left(2\right)+\mathrm{log}\left(x\right)-\mathrm{log}\left(3\right)\hfill\\ \end{align*}[/latex]
[latex]\\[/latex]
Finally we notice that [latex]x=-3[/latex] is not in the domain of the simplified expression either.

How To

Given the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.

  1. Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.
  2. Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.
  3. Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.

Example 10:  Using the Quotient Rule for Logarithms

Expand [latex]{\mathrm{log}}_{2}\left(\frac{15x\left(x-1\right)}{\left(3x+4\right)\left(2-x\right)}\right).[/latex]

Try it #10

Expand [latex]\mathrm{log}\left(\frac{7{x}^{2}+21x}{7x\left(x-1\right)\left(x-2\right)}\right).[/latex]

Power Rule for Logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as [latex]{x}^{2}?[/latex]  One method is as follows:

[latex]\begin{array}{ll}{\mathrm{log}}_{b}\left({x}^{2}\right)\hfill & ={\mathrm{log}}_{b}\left(x\cdot x\right)\hfill \\ \hfill & ={\mathrm{log}}_{b}\left(x\right)+{\mathrm{log}}_{b}\left(x\right)\hfill \\ \hfill & =2{\mathrm{log}}_{b}\left(x\right)\hfill \end{array}[/latex][latex]\\[/latex]

Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

[latex]\begin{array}{lll}100={10}^{2}\text{ }\text{ }\text{ }\text{ }\text{ }\hfill & \sqrt{3}={3}^{\frac{1}{2}}\text{ }\text{ }\text{ }\text{ }\text{ }\hfill & \frac{1}{e}={e}^{-1}\hfill \end{array}[/latex][latex]\\[/latex]

The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

[latex]{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}\left(M\right)[/latex]
[latex]\\[/latex]

How To

Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.

  1. Express the argument as a power, if needed.
  2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.

example 11:  Expanding a Logarithm with Powers

Expand [latex]{\mathrm{log}}_{2}\left({x}^{5}\right).[/latex]

Try it #11

Expand [latex]\mathrm{ln}\left({x}^{2}\right).[/latex]

example 12:  Rewriting an Expression as a Power before Using the Power Rule

Expand [latex]{\mathrm{log}}_{3}\left(25\right)[/latex] using the power rule for logs.

Try it #12

Expand [latex]\mathrm{ln}\left(\frac{1}{{x}^{2}}\right).[/latex]

 

Access this online resource for additional instruction and practice with logarithms.

Key Equations

Definition of the logarithmic function For [latex]x>0,b>0,b\ne 1,[/latex]

[latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] if and only if [latex]{b}^{y}=x.[/latex]
Definition of the common logarithm For [latex]x>0,[/latex] [latex]y=\mathrm{log}\left(x\right)[/latex] if and only if [latex]{10}^{y}=x.[/latex]
Definition of the natural logarithm For [latex]x>0,[/latex] [latex]y=\mathrm{ln}\left(x\right)[/latex] if and only if [latex]{e}^{y}=x.[/latex]
Properties of  logarithms [latex]{\mathrm{log}}_{b}\left({b}^{x}\right)=x[/latex]

[latex]{b}^{{\mathrm{log}}_{b}x}=x,\text{ }x>0[/latex]

[latex]{\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)[/latex] for [latex]b>0[/latex]

[latex]{\mathrm{log}}_{b}\left(\frac{M}{N}\right)={\mathrm{log}}_{b}\left(M\right)-{\mathrm{log}}_{b}\left(N\right)[/latex]

[latex]{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}\left(M\right)[/latex]

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.
  • Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm.
  • Logarithmic functions with base [latex]b[/latex] can be evaluated mentally using previous knowledge of powers of [latex]b.[/latex]
  • Common logarithms can be evaluated mentally using previous knowledge of powers of [latex]10.[/latex]
  • When common logarithms cannot be evaluated mentally, a calculator can be used.
  • Real-world exponential problems with base [latex]10[/latex] can be rewritten as a common logarithm and then evaluated using a calculator.
  • Natural logarithms can be evaluated using a calculator.
  • Properties of logarithms can be used to simplify expressions and expand them into sums and differences.

Glossary

common logarithm
the exponent to which 10 must be raised to get [latex]x;[/latex] [latex]{\mathrm{log}}_{10}\left(x\right)[/latex] is written simply as [latex]\mathrm{log}\left(x\right).[/latex]
logarithm
the exponent to which [latex]b[/latex] must be raised to get [latex]x;[/latex] written [latex]y={\mathrm{log}}_{b}\left(x\right).[/latex]
natural logarithm
the exponent to which the number [latex]e[/latex] must be raised to get [latex]x;[/latex] [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] is written as [latex]\mathrm{ln}\left(x\right).[/latex]
  1. http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013. ↵
  2. http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013. ↵
  3. http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. ↵
  4. http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013. ↵