Skip to Content
OpenStax Logo
Precalculus

4.3 Logarithmic Functions

Precalculus4.3 Logarithmic Functions
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

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.
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 homes19. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,20 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 Scale21 whereas the Japanese earthquake registered a 9.0.22

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 84 = 10 4 =10,000 10 84 = 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, 10 x =500, where x x represents the difference in magnitudes on the Richter Scale. How would we solve for x? 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. 10 x =500. We know that 10 2 =100 10 2 =100 and 10 3 =1000, 10 3 =1000, so it is clear that x x must be some value between 2 and 3, since y= 10 x y= 10 x 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 y= b x y= b x is one-to-one, so its inverse, x= b y x= b y is also a function. As is the case with all inverse functions, we simply interchange x x and y y and solve for y y to find the inverse function. To represent y y as a function of x, x, we use a logarithmic function of the form y= log b ( x ). y= log b ( x ). The base b b logarithm of a number is the exponent by which we must raise b b to get that number.

We read a logarithmic expression as, “The logarithm with base b b of x x is equal to y, y, ” or, simplified, “log base b b of x x is y. y. ” We can also say, “ b b raised to the power of y y is x, 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, 2 5 =32, we can write log 2 32=5. 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 b y =x, b>0,b1 log b ( x )=y b y =x, b>0,b1

Note that the base b b is always positive.

Because logarithm is a function, it is most correctly written as log b (x), log b (x), using parentheses to denote function evaluation, just as we would with f(x). 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. log b x. Note that many calculators require parentheses around the x. 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 ) y= log b ( x ) and y= b x y= b x are inverse functions.

Definition of the Logarithmic Function

A logarithm base b b of a positive number x x satisfies the following definition.

For x>0,b>0,b1, x>0,b>0,b1,

y= log b ( x ) is equivalent to  b y =x y= log b ( x ) is equivalent to  b y =x

where,

  • we read log b ( x ) log b ( x ) as, “the logarithm with base b b of x x ” or the “log base b b of x." x."
  • the logarithm y y is the exponent to which b b must be raised to get x. x.

Also, since the logarithmic and exponential functions switch the x x and y y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base b is (0,). b is (0,).
  • the range of the logarithm function with base b is (,). b is (,).

Q&A

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 log b ( x )=y, log b ( x )=y, convert it to exponential form.

  1. Examine the equation y= log b (x) y= log b (x) and identify b,y,andx. b,y,andx.
  2. Rewrite log b (x)=y log b (x)=y as b y =x. b y =x.

Example 1

Converting from Logarithmic Form to Exponential Form

Write the following logarithmic equations in exponential form.

  1. log 6 ( 6 )= 1 2 log 6 ( 6 )= 1 2
  2. log 3 ( 9 )=2 log 3 ( 9 )=2
Try It #1

Write the following logarithmic equations in exponential form.

  1. log 10 ( 1,000,000 )=6 log 10 ( 1,000,000 )=6
  2. log 5 ( 25 )=2 log 5 ( 25 )=2

Converting from Exponential to Logarithmic Form

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

Example 2

Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

  1. 2 3 =8 2 3 =8
  2. 5 2 =25 5 2 =25
  3. 10 4 = 1 10,000 10 4 = 1 10,000
Try It #2

Write the following exponential equations in logarithmic form.

  1. 3 2 =9 3 2 =9
  2. 5 3 =125 5 3 =125
  3. 2 1 = 1 2 2 1 = 1 2

Evaluating Logarithms

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

Now consider solving log 7 49 log 7 49 and log 3 27 log 3 27 mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know 7 2 =49. 7 2 =49. Therefore, log 7 49=2 log 7 49=2
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know 3 3 =27. 3 3 =27. Therefore, log 3 27=3 log 3 27=3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log 2 3 4 9 log 2 3 4 9 mentally.

  • We ask, “To what exponent must 2 3 2 3 be raised in order to get 4 9 ? 4 9 ? ” We know 2 2 =4 2 2 =4 and 3 2 =9, 3 2 =9, so ( 2 3 ) 2 = 4 9 . ( 2 3 ) 2 = 4 9 . Therefore, log 2 3 ( 4 9 )=2. log 2 3 ( 4 9 )=2.

How To

Given a logarithm of the form y= log b ( x ), y= log b ( x ), evaluate it mentally.

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

Example 3

Solving Logarithms Mentally

Solve y= log 4 ( 64 ) y= log 4 ( 64 ) without using a calculator.

Try It #3

Solve y= log 121 ( 11 ) y= log 121 ( 11 ) without using a calculator.

Example 4

Evaluating the Logarithm of a Reciprocal

Evaluate y= log 3 ( 1 27 ) y= log 3 ( 1 27 ) without using a calculator.

Try It #4

Evaluate y= log 2 ( 1 32 ) y= log 2 ( 1 32 ) 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 log( x ) log( x ) means log 10 ( x ). log 10 ( x ). 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 10. 10. We write log 10 ( x ) log 10 ( x ) simply as log( x ). log( x ). The common logarithm of a positive number x x satisfies the following definition.

For x>0, x>0,

y=log( x ) is equivalent to  10 y =x y=log( x ) is equivalent to  10 y =x

We read log( x ) log( x ) as, “the logarithm with base 10 10 of x x ” or “log base 10 of x. x.

The logarithm y y is the exponent to which 10 10 must be raised to get x. x.

How To

Given a common logarithm of the form y=log( x ), y=log( x ), evaluate it mentally.

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

Example 5

Finding the Value of a Common Logarithm Mentally

Evaluate y=log(1000) y=log(1000) without using a calculator.

Try It #5

Evaluate y=log(1,000,000). y=log(1,000,000).

How To

Given a common logarithm with the form y=log( x ), y=log( x ), evaluate it using a calculator.

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

Example 6

Finding the Value of a Common Logarithm Using a Calculator

Evaluate y=log( 321 ) y=log( 321 ) to four decimal places using a calculator.

Analysis

Note that 10 2 =100 10 2 =100 and that 10 3 =1000. 10 3 =1000. Since 321 is between 100 and 1000, we know that log( 321 ) log( 321 ) must be between log( 100 ) log( 100 ) and log( 1000 ). log( 1000 ). This gives us the following:

100 < 321 < 1000 2 < 2.5065 < 3 100 < 321 < 1000 2 < 2.5065 < 3
Try It #6

Evaluate y=log( 123 ) y=log( 123 ) 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 10 x =500 10 x =500 represents this situation, where x x 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 8,500 8,500 times greater than the amount of energy released from another. The equation 10 x =8500 10 x =8500 represents this situation, where x x 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 e. e. Base e e logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base e e logarithm, log e ( x ), log e ( x ), has its own notation, ln(x). ln(x).

Most values of ln( x ) 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. ln1=0. For other natural logarithms, we can use the ln ln key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e e using the inverse property of logarithms.

Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e. e. We write log e ( x ) log e ( x ) simply as ln( x ). ln( x ). The natural logarithm of a positive number x x satisfies the following definition.

For x>0, x>0,

y=ln( x ) is equivalent to  e y =x y=ln( x ) is equivalent to  e y =x

We read ln( x ) ln( x ) as, “the logarithm with base e e of x x ” or “the natural logarithm of x. x.

The logarithm y y is the exponent to which e e must be raised to get x. x.

Since the functions y=e x y=e x and y=ln( x ) y=ln( x ) are inverse functions, ln( e x )=x ln( e x )=x for all x x and e = ln(x) x e = ln(x) x for x>0. x>0.

How To

Given a natural logarithm with the form y=ln( x ), y=ln( x ), evaluate it using a calculator.

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

Example 8

Evaluating a Natural Logarithm Using a Calculator

Evaluate y=ln( 500 ) y=ln( 500 ) to four decimal places using a calculator.

Try It #8

Evaluate ln(−500). ln(−500).

Media

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

Footnotes

  • 19 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.
  • 20 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013.
  • 21 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013.
  • 22 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.
  • 23 http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014.

4.3 Section Exercises

Verbal

1.

What is a base b b logarithm? Discuss the meaning by interpreting each part of the equivalent equations b y =x b y =x and log b x=y log b x=y for b>0,b1. b>0,b1.

2.

How is the logarithmic function f(x)= log b x f(x)= log b x related to the exponential function g(x)= b x ? g(x)= b x ? What is the result of composing these two functions?

3.

How can the logarithmic equation log b x=y log b x=y be solved for x x using the properties of exponents?

4.

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

5.

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

Algebraic

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

6.

log 4 (q)=m log 4 (q)=m

7.

log a (b)=c log a (b)=c

8.

log 16 ( y )=x log 16 ( y )=x

9.

log x ( 64 )=y log x ( 64 )=y

10.

log y ( x )=−11 log y ( x )=−11

11.

log 15 ( a )=b log 15 ( a )=b

12.

log y ( 137 )=x log y ( 137 )=x

13.

log 13 ( 142 )=a log 13 ( 142 )=a

14.

log(v)=t log(v)=t

15.

ln(w)=n ln(w)=n

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

16.

4 x =y 4 x =y

17.

c d =k c d =k

18.

m 7 =n m 7 =n

19.

19 x =y 19 x =y

20.

x 10 13 =y x 10 13 =y

21.

n 4 =103 n 4 =103

22.

( 7 5 ) m =n ( 7 5 ) m =n

23.

y x = 39 100 y x = 39 100

24.

10 a =b 10 a =b

25.

e k =h e k =h

For the following exercises, solve for x x by converting the logarithmic equation to exponential form.

26.

log 3 (x)=2 log 3 (x)=2

27.

log 2 (x)=3 log 2 (x)=3

28.

log 5 (x)=2 log 5 (x)=2

29.

log 3 ( x )=3 log 3 ( x )=3

30.

log 2 (x)=6 log 2 (x)=6

31.

log 9 (x)= 1 2 log 9 (x)= 1 2

32.

log 18 (x)=2 log 18 (x)=2

33.

log 6 ( x )=3 log 6 ( x )=3

34.

log(x)=3 log(x)=3

35.

ln(x)=2 ln(x)=2

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

36.

log( 100 8 ) log( 100 8 )

37.

10 log(32) 10 log(32)

38.

2log(.0001) 2log(.0001)

39.

e ln( 1.06 ) e ln( 1.06 )

40.

ln( e 5.03 ) ln( e 5.03 )

41.

e ln( 10.125 ) +4 e ln( 10.125 ) +4

Numeric

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

42.

log 3 ( 1 27 ) log 3 ( 1 27 )

43.

log 6 ( 6 ) log 6 ( 6 )

44.

log 2 ( 1 8 )+4 log 2 ( 1 8 )+4

45.

6 log 8 (4) 6 log 8 (4)

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

46.

log(10,000) log(10,000)

47.

log(0.001) log(0.001)

48.

log(1)+7 log(1)+7

49.

2log( 100 3 ) 2log( 100 3 )

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

50.

ln( e 1 3 ) ln( e 1 3 )

51.

ln(1) ln(1)

52.

ln( e 0.225 )3 ln( e 0.225 )3

53.

25ln( e 2 5 ) 25ln( e 2 5 )

Technology

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

54.

log(0.04) log(0.04)

55.

ln(15) ln(15)

56.

ln( 4 5 ) ln( 4 5 )

57.

log( 2 ) log( 2 )

58.

ln( 2 ) ln( 2 )

Extensions

59.

Is x=0 x=0 in the domain of the function f(x)=log(x)? f(x)=log(x)? If so, what is the value of the function when x=0? x=0? Verify the result.

60.

Is f(x)=0 f(x)=0 in the range of the function f(x)=log(x)? f(x)=log(x)? If so, for what value of x? x? Verify the result.

61.

Is there a number x x such that lnx=2? lnx=2? If so, what is that number? Verify the result.

62.

Is the following true: log 3 (27) log 4 ( 1 64 ) =−1? log 3 (27) log 4 ( 1 64 ) =−1? Verify the result.

63.

Is the following true: ln( e 1.725 ) ln( 1 ) =1.725? ln( e 1.725 ) ln( 1 ) =1.725? Verify the result.

Real-World Applications

64.

The exposure index EI EI for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation EI= log 2 ( f 2 t ), EI= log 2 ( f 2 t ), where f f is the “f-stop” setting on the camera, and t t is the exposure time in seconds. Suppose the f-stop setting is 8 8 and the desired exposure time is 2 2 seconds. What will the resulting exposure index be?

65.

Refer to the previous exercise. Suppose the light meter on a camera indicates an EI EI of 2, 2, and the desired exposure time is 16 seconds. What should the f-stop setting be?

66.

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula log I 1 I 2 = M 1 M 2 log I 1 I 2 = M 1 M 2 where M M 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 9.0.23 How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
Citation information

© Feb 10, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.