Exponential and Logarithmic Functions

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 whererepresents 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 solveWe know thatand so it is clear thatmust be some value between 2 and 3, sinceis 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 functionis one-to-one, so its inverse,is also a function. As is the case with all inverse functions, we simply interchangeandand solve forto find the inverse function. To representas a function of we use a logarithmic function of the formThe baselogarithm of a number is the exponent by which we must raiseto get that number.

We read a logarithmic expression as, “The logarithm with baseofis equal to” or, simplified, “log baseofis” We can also say, “raised to the power ofis” 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 writeWe 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 baseis always positive.

Because logarithm is a function, it is most correctly written as using parentheses to denote function evaluation, just as we would withHowever, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, asNote 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 meansandare inverse functions.

Definition of the Logarithmic Function

A logarithm baseof a positive numbersatisfies the following definition.

For

where,

• we readas, “the logarithm with baseof” or the “log baseof
• the logarithmis the exponent to whichmust be raised to get

Also, since the logarithmic and exponential functions switch theandvalues, 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.

1. Examine the equationand identify
2. Rewriteas

Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.

First, identify the values ofThen, write the equation in the form

1. Here,Therefore, the equationis equivalent to

2. Here,Therefore, the equationis equivalent to

Try It

Write the following logarithmic equations in exponential form.

1. is equivalent to
2. 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 baseexponentand outputThen we write

Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

First, identify the values ofThen, write the equation in the form

1. Here,$\,x=3,$andTherefore, the equationis equivalent to

2. Here,$\,x=2,$andTherefore, the equationis equivalent to

3. Here,$\,x=-4,$andTherefore, the equationis equivalent to

Try It

Write the following exponential equations in logarithmic form.

1. is equivalent to
2. is equivalent to
3. is equivalent to

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, considerWe ask, “To what exponent must be raised in order to get 8?” Because we already know it follows that

Now consider solvingandmentally.

• We ask, “To what exponent must 7 be raised in order to get 49?” We knowTherefore,
• We ask, “To what exponent must 3 be raised in order to get 27?” We knowTherefore,

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluatementally.

• We ask, “To what exponent mustbe raised in order to get” We knowandsoTherefore,

How To

Given a logarithm of the formevaluate it mentally.

1. Rewrite the argumentas a power of${b}^{y}=x.\,$
2. Use previous knowledge of powers ofidentifyby asking, “To what exponent shouldbe raised in order to get

Solving Logarithms Mentally

Solvewithout using a calculator.

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

Solvewithout using a calculator.

(recalling that)

Evaluating the Logarithm of a Reciprocal

Evaluatewithout using a calculator.

First we rewrite the logarithm in exponential form:Next, we ask, “To what exponent must 3 be raised in order to get

We knowbut what must we do to get the reciprocal,Recall from working with exponents thatWe use this information to write

Try It

Evaluatewithout 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 expressionmeansWe 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 baseWe writesimply asThe common logarithm of a positive numbersatisfies the following definition.

For

We readas, “the logarithm with baseof” or “log base 10 of

The logarithmis the exponent to whichmust be raised to get

How To

Given a common logarithm of the form evaluate it mentally.

1. Rewrite the argumentas a power of${10}^{y}=x.$
2. Use previous knowledge of powers ofto identifyby asking, “To what exponent mustbe raised in order to get

Finding the Value of a Common Logarithm Mentally

Evaluatewithout using a calculator.

First we rewrite the logarithm in exponential form:Next, we ask, “To what exponent mustbe raised in order to get 1000?” We know

Evaluate

How To

Given a common logarithm with the formevaluate it using a calculator.

1. Press [LOG].
2. Enter the value given forfollowed by [ ) ].
3. Press [ENTER].

Finding the Value of a Common Logarithm Using a Calculator

Evaluateto four decimal places using a calculator.

• Press [LOG].
• Enter 321, followed by [ ) ].
• Press [ENTER].

Rounding to four decimal places,

Analysis

Note thatand thatSince 321 is between 100 and 1000, we know thatmust be betweenandThis gives us the following:

Try It

Evaluateto four decimal places using a calculator.

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 equationrepresents this situation, whereis the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

We begin by rewriting the exponential equation in logarithmic form.

Next we evaluate the logarithm using a calculator:

• Press [LOG].
• Enterfollowed by [ ) ].
• Press [ENTER].
• To the nearest thousandth,

Try It

The amount of energy released from one earthquake wastimes greater than the amount of energy released from another. The equationrepresents this situation, whereis the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

The difference in magnitudes was about

Using Natural Logarithms

The most frequently used base for logarithms isBaselogarithms are important in calculus and some scientific applications; they are called natural logarithms. The baselogarithm, has its own notation,

Most values ofcan 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 thekey that can be found on most scientific calculators. We can also find the natural logarithm of any power ofusing 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 readas, “the logarithm with baseof” or “the natural logarithm of

The logarithmis the exponent to whichmust be raised to get

Since the functionsandare inverse functions,for allandfor

How To

Given a natural logarithm with the form evaluate it using a calculator.

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

Evaluating a Natural Logarithm Using a Calculator

Evaluateto four decimal places using a calculator.

• Press [LN].
• Enterfollowed by [ ) ].
• Press [ENTER].

Rounding to four decimal places,

Try It

Evaluate

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 Definition of the common logarithm For$y=\mathrm{log}\left(x\right)\text{ }$if and only if Definition of the natural logarithm For$y=\mathrm{ln}\left(x\right)\text{ }$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 basecan be evaluated mentally using previous knowledge of powers ofSee (Figure) and (Figure).
• Common logarithms can be evaluated mentally using previous knowledge of powers ofSee (Figure).
• When common logarithms cannot be evaluated mentally, a calculator can be used. See (Figure).
• Real-world exponential problems with basecan 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

What is a baselogarithm? Discuss the meaning by interpreting each part of the equivalent equationsandfor

[hidden-answer a=”fs-id1165137411524″]A logarithm is an exponent. Specifically, it is the exponent to which a baseis raised to produce a given value. In the expressions given, the basehas the same value. The exponent,in the expressioncan also be written as the logarithm,and the value ofis the result of raisingto the power of

How is the logarithmic functionrelated to the exponential functionWhat is the result of composing these two functions?

How can the logarithmic equationbe solved forusing the properties of exponents?

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

Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base and how does the notation differ?

Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base and how does the notation differ?

The natural logarithm is a special case of the logarithm with basein that the natural log always has baseRather than notating the natural logarithm asthe notation used is

Algebraic

For the following exercises, rewrite each equation in exponential form.

For the following exercises, rewrite each equation in logarithmic form.

For the following exercises, solve forby converting the logarithmic equation to exponential form.

For the following exercises, use the definition of common and natural logarithms to simplify.

Numeric

For the following exercises, evaluate the baselogarithmic expression without using a calculator.

For the following exercises, evaluate the common logarithmic expression without using a calculator.

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

Extensions

Isin the domain of the functionIf so, what is the value of the function whenVerify the result.

No, the function has no defined value forTo verify, supposeis in the domain of the functionThen there is some numbersuch thatRewriting as an exponential equation gives: which is impossible since no such real numberexists. Therefore,is not the domain of the function

Isin the range of the functionIf so, for what value ofVerify the result.

Is there a numbersuch thatIf so, what is that number? Verify the result.

Yes. Suppose there exists a real numbersuch thatRewriting as an exponential equation gives which is a real number. To verify, letThen, by definition,

Is the following true:Verify the result.

Is the following true:Verify the result.

No; sois undefined.

Real-World Applications

The exposure indexfor a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation whereis the “f-stop” setting on the camera, and is the exposure time in seconds. Suppose the f-stop setting isand the desired exposure time isseconds. What will the resulting exposure index be?

Refer to the previous exercise. Suppose the light meter on a camera indicates anof and the desired exposure time is 16 seconds. What should the f-stop setting be?

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formulawhereis 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$\,{\mathrm{log}}_{10}\left(x\right)\,$ is written simply as
logarithm
the exponent to whichmust be raised to getwritten
natural logarithm
the exponent to which the numbermust be raised to get$\,{\mathrm{log}}_{e}\left(x\right)\,$is written as