Chapter 1: Algebra Review

# Section 1.8: Logarithmic Functions

Learning Objectives

By the end of this section, the student should be able to:

- Convert from logarithmic to exponential form.
- Convert from exponential to logarithmic form.
- Evaluate logarithms.
- Use common logarithms.
- Use natural logarithms.

# Logarithm Basics

Logarithms are the inverse of exponential functions – they allow us to undo exponential functions and solve for the exponent. They are also commonly used to express quantities that vary widely in size.

#### Logarithm Equivalent to an Exponential

The logarithm (base [latex]b[/latex]) function, written [latex]\log_b (x)[/latex], is the inverse of the exponential function (base [latex]b[/latex]), [latex]b^x[/latex].

This means the statement [latex]b^a=c[/latex] is equivalent to the statement [latex]\log_b (c)=a[/latex].

#### Properties of Logs: Inverse Properties

- [latex]\log_b(b^x)=x[/latex]
- [latex]b^{\log_b(x)}=x[/latex]

Video

#### Example 1

Write these exponential equations as logarithmic equations:

- [latex]2^3=8[/latex]
- [latex]5^2=25[/latex]
- [latex]10^{-4}=\frac{1}{10000}[/latex]
- [latex]2^3=8[/latex] is equivalent to [latex]\log_2(8)=3[/latex].
- [latex]5^2=25[/latex] is equivalent to [latex]\log_5(25)=2[/latex].
- [latex]10^{-4}=\frac{1}{10000}[/latex] is equivalent to [latex]\log_{10}\left(\frac{1}{10000}\right)=-4[/latex].

#### Example 2

Solve [latex]2^x=10[/latex] for [latex]x[/latex].

By rewriting this expression as a logarithm, we get [latex]x=\log_2(10)[/latex].

Video

#### Properties of Logs: Inverse Properties

The **common log** is the logarithm with base 10 and is typically written [latex]\log(x)[/latex].

The **natural log** is the logarithm with base [latex]e[/latex] and is typically written [latex]\ln(x)[/latex].

#### Example 3

Evaluate [latex]\log(1000)[/latex] using the definition of the common log.

To evaluate [latex]\log(1000)[/latex], we can say [latex]x=\log(1000)[/latex], then rewrite into exponential form using the common log base of 10:[latex]10^x=1000.[/latex]

From this, we might recognize that 1000 is the cube of 10, so [latex]x = 3[/latex].

We also can use the inverse property of logs to write [latex]log_{10}\left(10^3\right) =3[/latex].

Number | Number as exponential | log(number) |

[latex]1000[/latex] | [latex]10^3[/latex] | [latex]3[/latex] |

[latex]100[/latex] | [latex]10^2[/latex] | [latex]2[/latex] |

[latex]10[/latex] | [latex]10^1[/latex] | [latex]1[/latex] |

[latex]1[/latex] | [latex]10^0[/latex] | [latex]0[/latex] |

[latex]0.1[/latex] | [latex]10^{-1}[/latex] | [latex]-1[/latex] |

[latex]0.01[/latex] | [latex]10^{-2}[/latex] | [latex]-2[/latex] |

[latex]0.001[/latex] | [latex]10^{-3}[/latex] | [latex]-3[/latex] |

#### Example 4

Evaluate [latex]\log(500)[/latex] using your calculator or computer.

Using a computer or calculator, we can evaluate and find that [latex]\log(500)\approx 2.69897[/latex].

Another property provides the basis for solving exponential equations.

#### Properties of Logs: Exponential Property

[latex]\log_b\left(A^r\right)=r\,\log_b(A)[/latex]

#### Solving Exponential Equations

- Isolate the exponential expressions when possible.
- Take the logarithm of both sides.
- Utilize the exponent property for logarithms to pull the variable out of the exponent.
- Use algebra to solve for the variable.

Video

#### Example 5

In the last section, we predicted the population (in billions) of India [latex]t[/latex] years after [latex]2008[/latex] by using the function [latex]f(t)=1.14(1+0.0134)^t[/latex]. If the population continues following this trend, when will the population reach [latex]2[/latex] billion?

We need to solve for the [latex]t[/latex] so that [latex]f(t) = 2[/latex].

[latex]2=1.14(1.0134)^t[/latex] | Initial equation. |

[latex]\dfrac{2}{1.14}=1.0134^t[/latex] | Divide by 1.14 to isolate the exponential expression. |

[latex]\ln\left(\dfrac{2}{1.14}\right)=\ln\left(1.0134^t\right)[/latex] | Take the logarithm of both sides of the equation. |

[latex]\ln\left(\dfrac{2}{1.14}\right)=t\,\ln(1.0134)[/latex] | Apply the exponent property on the right side. |

[latex]t = \dfrac{\ln\left(\dfrac{2}{1.14}\right)}{\ln(1.0134)}[/latex] | Divide both sides by [latex]\ln(1.0134)[/latex] |

[latex]t\approx 42.23 \text{ years}[/latex] |

If this growth rate continues, the model predicts the population of India will reach [latex]2[/latex] billion about [latex]42[/latex] years after [latex]2008[/latex], or approximately in the year [latex]2050[/latex].

#### Example 6

Solve [latex]5e^{-0.3t}=2[/latex] for [latex]t[/latex].

First we divide by 5 to isolate the exponential: [latex]e^{-0.3t}=\frac{2}{5}.[/latex]

Since this equation involves [latex]e[/latex], it makes sense to use the natural log:

[latex]\ln\left(e^{-0.3t}\right)=\ln\left(\dfrac{2}{5}\right)[/latex] | Taking the natural log of both sides. |

[latex]-0.3t=\ln\left(\dfrac{2}{5}\right)[/latex] | Utilizing the inverse property for logs. |

[latex]t = \dfrac{\ln\left(\dfrac{2}{5}\right)}{-0.3}[/latex] | Now dividing by -0.3. |

[latex]t\approx 3.054[/latex] |

In addition to solving exponential equations, logarithmic expressions are common in many physical situations.

#### Example 7

In chemistry, [latex]pH[/latex] is a measure of the acidity or basicity of a liquid. The [latex]pH[/latex] is related to the concentration of hydrogen ions, [latex]\left[H^+\right][/latex], measured in moles per liter, by the equation[latex]\text{pH}=-\log\left(\left[H^+\right]\right).[/latex]

If a liquid has concentration of [latex]0.0001[/latex] moles per liter, determine the [latex]pH[/latex]. Determine the hydrogen ion concentration of a liquid with [latex]pH[/latex] of [latex]7[/latex].

To answer the first question, we evaluate the expression [latex]-\log(0.0001)[/latex]. While we could use our calculators for this, we do not really need them here, since we can use the inverse property of logs: [latex]-\log(0.0001)=-\log\left(10^{-4}\right)=-(-4)=4.[/latex]

To answer the second question, we need to solve the equation [latex]7=-\log\left(\left[H^+\right]\right)[/latex]. Begin by isolating the logarithm on one side of the equation by multiplying both sides by -1: [latex]-7=\log\left(\left[H^+\right]\right)[/latex]. Rewriting into exponential form yields the answer: [latex]\left[H^+\right]=10^{-7}=0.0000001\text{ moles per liter}.[/latex]

# Logarithm Graphs

#### Graphical Features of the Logarithm

Graphically, given the function [latex]g(x)=\log_b(x)[/latex].

- The graph has a horizontal intercept at (1, 0).
- The graph has a vertical asymptote at [latex]x = 0[/latex].
- The graph is increasing and concave down.
- The domain of the function is [latex]x \gt 0[/latex], or [latex](0, \infty)[/latex] in interval notation.
- The range of the function is all real numbers, or [latex](-\infty, \infty)[/latex] in interval notation.

When sketching a general logarithm with base [latex]b[/latex], it can be helpful to remember that the graph will pass through the points [latex]\left(\frac{1}{b}, -1\right)[/latex], [latex](1, 0)[/latex], and [latex](b, 1)[/latex].

To get a feeling for how the base affects the shape of the graph, examine the graphs below:

Another important observation made was the domain of the logarithm: [latex]x \gt 0[/latex]. Like the reciprocal and square root functions, the logarithm has a restricted domain that must be considered when finding the domain of a composition involving a log.

#### Example 8

Find the domain of the function [latex]f(x)=\log(5-2x)[/latex].

The logarithm is only defined when the input is positive, so this function will only be defined when [latex]5-2x \gt 0[/latex]. Solving this inequality, [latex]-2x \gt -5[/latex], so [latex]x\lt \frac{5}{2}[/latex].

The domain of this function is [latex]x\lt \frac{5}{2}[/latex], or, in interval notation, [latex]\left(-\infty, \frac{5}{2} \right)[/latex].

# Changing Logarithm Bases

[Content section *Introduction from Calculus, Volume 1, by Strang and Herman, OpenStax (Web), licensed under a **CC BY-NC-SA 4.0 License.*]

When evaluating a logarithmic function with a calculator, you may have noticed that the only options are [latex]\log_{10}[/latex] or log, called the *common logarithm*, or ln, which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base [latex]b[/latex]. If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.

### Rule: Change-of-Base Formulas

Let [latex]a \lt 0, \, b \lt 0[/latex], and [latex]a\ne 1, \, b\ne 1[/latex].

- [latex]a^x=b^{x \log_b a}[/latex] for any real number [latex]x[/latex].

If [latex]b=e[/latex], this equation reduces to [latex]a^x=e^{x \log_e a}=e^{x \ln a}[/latex]. - [latex]\log_a x=\frac{\log_b x}{\log_b a}[/latex] for any real number [latex]x \lt 0[/latex].

If [latex]b=e[/latex], this equation reduces to [latex]\log_a x=\frac{\ln x}{\ln a}[/latex].

### Example 9

Use a calculating utility to evaluate [latex]\log_3 7[/latex] with the change-of-base formula presented earlier.

## Solution

Use the second equation with [latex]a=3[/latex] and [latex]e=3[/latex]:

[latex]\log_3 7=\frac{\ln 7}{\ln 3} \approx 1.77124[/latex].

### Example 10

Use the change-of-base formula and a calculating utility to evaluate [latex]\log_4 6[/latex].

#### Hint

Use the change of base to rewrite this expression in terms of expressions involving the natural logarithm function.

## Solution

1.29248

### Chapter Opener: The Richter Scale for Earthquakes

*From Calculus, Volume 1, by Strang and Herman, OpenStax (Web), licensed under a **CC BY-NC-SA 4.0 License.*

In 1935, Charles Richter developed a scale (now known as the *Richter scale*) to measure the magnitude of an earthquake. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude [latex]R_1[/latex] on the Richter scale and a second earthquake with magnitude [latex]R_2[/latex] on the Richter scale. Suppose [latex]R_1 R_2[/latex], which means the earthquake of magnitude [latex]R_1[/latex] is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If [latex]A_1[/latex] is the amplitude measured for the first earthquake and [latex]A_2[/latex] is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:

Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,

Therefore,

which implies [latex]A_1 / A_2 = 10[/latex] or [latex]A_1 = 10A_2[/latex]. Since [latex]A_1[/latex] is 10 times the size of [latex]A_2[/latex], we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation

Therefore, [latex]A_1=100A_2[/latex]. That is, the first earthquake is 100 times more intense than the second earthquake.

How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?

## Solution

To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:

[latex]9-7.3=\log_{10}(\frac{A_1}{A_2})[/latex].

Therefore, [latex]A_1 / A_2=10^{1.7}[/latex], and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.

Compare the relative severity of a magnitude 8.4 earthquake with a magnitude 7.4 earthquake.

#### Hint

[latex]R_1-R_2=\log_{10}(A_1 / A_2)[/latex].

## Solution

The magnitude 8.4 earthquake is roughly 10 times as severe as the magnitude 7.4 earthquake.

# Vocabulary

the exponent to which ** b** must be raised to get

**written by**

*x*

**y = log base b of x**the exponent to which 10 must be raised to get ** x**; log base 10 of x is written as

*.*

**log (x)**the exponent to which the number ** e** must be raised to get

**;**

*x***is written as**

*log base e of x***.**

*ln (x)*Horizontal intercept (x-intercept) of a line: The point where the line intersects the horizontal axis (x-axis) on a graph.

The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.

The domain of a function is the set of all possible input values for which the function is defined.

The range of a function is the set of all possible output values that the function can produce.