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 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 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 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 one-to-one, 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
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Here, and Therefore, the equation is equivalent to
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Here, and Therefore, the equation is equivalent to
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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.
Show Solution
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 means 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 of the Common Logarithm
A common logarithm is a logarithm with base We write simply as The common logarithm of a positive number satisfies the following definition.
For
We read as, “the logarithm with base of ” or “log base 10 of ”
The logarithm is the exponent to which must be raised to get
How To
Given a common logarithm of the form evaluate it mentally.
- Rewrite the argument as a power of
- Use previous knowledge of powers of to identify by asking, “To what exponent must be raised in order to get ”
Finding the Value of a Common Logarithm Mentally
Evaluate without using a calculator.
Show Solution
First we rewrite the logarithm in exponential form: Next, we ask, “To what exponent must be raised in order to get 1000?” We know
Therefore,
Try It
Evaluate
Show Solution
How To
Given a common logarithm with the form evaluate it using a calculator.
- Press [LOG].
- Enter the value given for followed by [ ) ].
- Press [ENTER].
Finding the Value of a Common Logarithm Using a Calculator
Evaluate to four decimal places using a calculator.
Show Solution
- Press [LOG].
- Enter 321, followed by [ ) ].
- Press [ENTER].
Rounding to four decimal places,
Analysis
Note that and that Since 321 is between 100 and 1000, we know that must be between and This gives us the following:
Try It
Evaluate to four decimal places using a calculator.
Show Solution
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 represents this situation, where is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
Show Solution
We begin by rewriting the exponential equation in logarithmic form.
Next we evaluate the logarithm using a calculator:
- Press [LOG].
- Enter followed by [ ) ].
- Press [ENTER].
- To the nearest thousandth,
The difference in magnitudes was about
Try It
The amount of energy released from one earthquake was times greater than the amount of energy released from another. The equation represents this situation, where is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
Show Solution
The difference in magnitudes was about
Using Natural Logarithms
The most frequently used base for logarithms is Base logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base logarithm, has its own notation,
Most values of can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, For other natural logarithms, we can use the key that can be found on most scientific calculators. We can also find the natural logarithm of any power of using the inverse property of logarithms.
Definition of the Natural Logarithm
A natural logarithm is a logarithm with base We write simply as The natural logarithm of a positive number satisfies the following definition.
For
We read as, “the logarithm with base of ” or “the natural logarithm of ”
The logarithm is the exponent to which must be raised to get
Since the functions and are inverse functions, for all and for
How To
Given a natural logarithm with the form evaluate it using a calculator.
- Press [LN].
- Enter the value given for followed by [ ) ].
- Press [ENTER].
Evaluating a Natural Logarithm Using a Calculator
Evaluate to four decimal places using a calculator.
Show Solution
- Press [LN].
- Enter followed by [ ) ].
- Press [ENTER].
Rounding to four decimal places,
Try It
Evaluate
Show Solution
It is not possible to take the logarithm of a negative number in the set of real numbers.
Access this online resource for additional instruction and practice with logarithms.
Key Equations
Definition of the logarithmic function | For < if and only if </td> |
Definition of the common logarithm | For if and only if |
Definition of the natural logarithm | For if and only if |
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 (Figure).
- Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See (Figure).
- Logarithmic functions with base can be evaluated mentally using previous knowledge of powers of See (Figure) and (Figure).
- Common logarithms can be evaluated mentally using previous knowledge of powers of See (Figure).
- When common logarithms cannot be evaluated mentally, a calculator can be used. See (Figure).
- Real-world exponential problems with base can be rewritten as a common logarithm and then evaluated using a calculator. See (Figure).
- Natural logarithms can be evaluated using a calculator (Figure).
Section Exercises
Verbal
1. What is a base logarithm? Discuss the meaning by interpreting each part of the equivalent equations and for
Show Solution
A logarithm is an exponent. Specifically, it is the exponent to which a base is raised to produce a given value. In the expressions given, the base has the same value. The exponent, in the expression can also be written as the logarithm, and the value of is the result of raising to the power of
2. How is the logarithmic function related to the exponential function What is the result of composing these two functions?
3. How can the logarithmic equation be solved for using the properties of exponents?
Show Solution
Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation and then properties of exponents can be applied to solve for
4. Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base and how does the notation differ?
5. Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base and how does the notation differ?
Show Solution
The natural logarithm is a special case of the logarithm with base in that the natural log always has base Rather than notating the natural logarithm as the notation used is
Algebraic
For the following exercises, rewrite each equation in exponential form.
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For the following exercises, rewrite each equation in logarithmic form.
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For the following exercises, solve for by converting the logarithmic equation to exponential form.
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For the following exercises, use the definition of common and natural logarithms to simplify.
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Numeric
For the following exercises, evaluate the base logarithmic expression without using a calculator.
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For the following exercises, evaluate the common logarithmic expression without using a calculator.
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For the following exercises, evaluate the natural logarithmic expression without using a calculator.
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Technology
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.
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Extensions
59. Is in the domain of the function If so, what is the value of the function when Verify the result.
Show Solution
No, the function has no defined value for To verify, suppose is in the domain of the function Then there is some number such that Rewriting as an exponential equation gives: which is impossible since no such real number exists. Therefore, is not the domain of the function
60. Is in the range of the function If so, for what value of Verify the result.
61. Is there a number such that If so, what is that number? Verify the result.
Show Solution
Yes. Suppose there exists a real number such that Rewriting as an exponential equation gives which is a real number. To verify, let Then, by definition,
62. Is the following true: Verify the result.
63. Is the following true: Verify the result.
Show Solution
No; so is undefined.
Real-World Applications
64. The exposure index for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation where is the “f-stop” setting on the camera, and is the exposure time in seconds. Suppose the f-stop setting is and the desired exposure time is seconds. What will the resulting exposure index be?
65. Refer to the previous exercise. Suppose the light meter on a camera indicates an of and the desired exposure time is 16 seconds. What should the f-stop setting be?
Show Solution
66. The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula where is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.[5] How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.
Glossary
- common logarithm
- the exponent to which 10 must be raised to get is written simply as
- logarithm
- the exponent to which must be raised to get written
- natural logarithm
- the exponent to which the number must be raised to get is written as
- http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013. ↵
- http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013. ↵
- http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. ↵
- http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. ↵
- http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014. ↵