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Five-Minute Check (over Lesson 3-1) Then/Now New Vocabulary Key Concept: Relating Logarithmic and Exponential Forms Example 1: Evaluate Logarithms Key Concept: Basic Properties of Logarithms Example 2: Apply Properties of Logarithms Key Concept: Basic Properties of Common Logarithms Example 3: Common Logarithms Key Concept: Basic Properties of Natural Logarithms Example 4: Natural Logarithms Example 5: Graphs of Logarithmic Functions Key Concept: Properties of Logarithmic Functions Example 6: Graph Transformations of Logarithmic Functions Example 7: Real-World Example: Use Logarithmic Functions Lesson Menu

A. Sketch the graph of f (x) = 3x + 1. B. C. D. 5–Minute Check 1

A. Sketch the graph of f (x) = 3x + 1. B. C. D. 5–Minute Check 1

B. Analyze the graph of f (x) = 3x + 1 B. Analyze the graph of f (x) = 3x + 1. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. A. D = (–∞, ∞); R = (1, ∞), y-intercept: 0; asymptote: y = –1; increasing (–∞, ∞) B. D = (–∞, ∞); R = (1, ∞), y-intercept: 2; asymptote: y = 1; ; increasing (–∞, ∞) C. D = (–∞, ∞); R = (0, ∞), y-intercept: 3; asymptote: y = 0; increasing (–∞, ∞) D. D = (–∞, ∞); R = (0, ∞), y-intercept: ; asymptote: y = 0; increasing (–∞, ∞) 5–Minute Check 1

B. Analyze the graph of f (x) = 3x + 1 B. Analyze the graph of f (x) = 3x + 1. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. A. D = (–∞, ∞); R = (1, ∞), y-intercept: 0; asymptote: y = –1; increasing (–∞, ∞) B. D = (–∞, ∞); R = (1, ∞), y-intercept: 2; asymptote: y = 1; ; increasing (–∞, ∞) C. D = (–∞, ∞); R = (0, ∞), y-intercept: 3; asymptote: y = 0; increasing (–∞, ∞) D. D = (–∞, ∞); R = (0, ∞), y-intercept: ; asymptote: y = 0; increasing (–∞, ∞) 5–Minute Check 1

The Barfields are saving for their son’s college education The Barfields are saving for their son’s college education. If they deposit $31,500 in an account earning 7.6% per year compounded continuously, how much will be in the account when their son goes to college in 9 years if there are no other deposits or withdrawals? A. $60,900.52 B. $62,291.85 C. $62,426.36 D. $29,436,407.74 5–Minute Check 2

The Barfields are saving for their son’s college education The Barfields are saving for their son’s college education. If they deposit $31,500 in an account earning 7.6% per year compounded continuously, how much will be in the account when their son goes to college in 9 years if there are no other deposits or withdrawals? A. $60,900.52 B. $62,291.85 C. $62,426.36 D. $29,436,407.74 5–Minute Check 2

You graphed and analyzed exponential functions. (Lesson 3-1) Evaluate expressions involving logarithms. Sketch and analyze graphs of logarithmic functions. Then/Now

logarithmic function with base b logarithm common logarithm natural logarithm Vocabulary

Key Concept 1

2y = 16 Write in exponential form. 2y = 24 16 = 24 Evaluate Logarithms A. Evaluate log216. log216 = y Let log216 = y. 2y = 16 Write in exponential form. 2y = 24 16 = 24 y = 4 Equality property of exponents. Answer: Example 1

2y = 16 Write in exponential form. 2y = 24 16 = 24 Evaluate Logarithms A. Evaluate log216. log216 = y Let log216 = y. 2y = 16 Write in exponential form. 2y = 24 16 = 24 y = 4 Equality property of exponents. Answer: 4 Example 1

5y = Write in exponential form. Evaluate Logarithms B. Evaluate . = y Let = y. 5y = Write in exponential form. 5y = 5–3 = 5–3 y = –3 Equality property of exponents. Answer: Example 1

5y = Write in exponential form. Evaluate Logarithms B. Evaluate . = y Let = y. 5y = Write in exponential form. 5y = 5–3 = 5–3 y = –3 Equality property of exponents. Answer: –3 Example 1

3y = Write in exponential form. Evaluate Logarithms C. Evaluate . = y Let log3 = y. 3y = Write in exponential form. 3y = 3–3 = 3–3 y = –3 Equality property of exponents. Answer: Example 1

3y = Write in exponential form. Evaluate Logarithms C. Evaluate . = y Let log3 = y. 3y = Write in exponential form. 3y = 3–3 = 3–3 y = –3 Equality property of exponents. Answer: –3 Example 1

17y = 17 Write in exponential form. 17y = 171 17= 171 Evaluate Logarithms D. Evaluate log17 17. log1717 = y Let log1717 = y. 17y = 17 Write in exponential form. 17y = 171 17= 171 y = 17 Equality property of exponents. Answer: Example 1

17y = 17 Write in exponential form. 17y = 171 17= 171 Evaluate Logarithms D. Evaluate log17 17. log1717 = y Let log1717 = y. 17y = 17 Write in exponential form. 17y = 171 17= 171 y = 17 Equality property of exponents. Answer: 1 Example 1

Evaluate . A. –4 B. 4 C. –2 D. 2 Example 1

Evaluate . A. –4 B. 4 C. –2 D. 2 Example 1

Key Concept 2

A. Evaluate log8 512. log8512 = log883 83 = 512 =3 logbbx = x Answer: Apply Properties of Logarithms A. Evaluate log8 512. log8512 = log883 83 = 512 =3 logbbx = x Answer: Example 2

A. Evaluate log8 512. log8512 = log883 83 = 512 =3 logbbx = x Apply Properties of Logarithms A. Evaluate log8 512. log8512 = log883 83 = 512 =3 logbbx = x Answer: 3 Example 2

B. Evaluate 22log22 15.2. 22log22 15.2 =15.2 blogbx = x Answer: Apply Properties of Logarithms B. Evaluate 22log22 15.2. 22log22 15.2 =15.2 blogbx = x Answer: Example 2

B. Evaluate 22log22 15.2. 22log22 15.2 =15.2 blogbx = x Answer: 15.2 Apply Properties of Logarithms B. Evaluate 22log22 15.2. 22log22 15.2 =15.2 blogbx = x Answer: 15.2 Example 2

Evaluate 7log7 4. A. 4 B. 7 C. 47 D. 74 Example 2

Evaluate 7log7 4. A. 4 B. 7 C. 47 D. 74 Example 2

Key Concept 3

A. Evaluate log 10,000. log10,000 = log104 10,000 = 104 = 4 log10x = x Common Logarithms A. Evaluate log 10,000. log10,000 = log104 10,000 = 104 = 4 log10x = x Answer: Example 3

A. Evaluate log 10,000. log10,000 = log104 10,000 = 104 = 4 log10x = x Common Logarithms A. Evaluate log 10,000. log10,000 = log104 10,000 = 104 = 4 log10x = x Answer: 4 Example 3

B. Evaluate 10log 12. 10log 12 = 12 10log x = x Answer: Common Logarithms B. Evaluate 10log 12. 10log 12 = 12 10log x = x Answer: Example 3

B. Evaluate 10log 12. 10log 12 = 12 10log x = x Answer: 12 Common Logarithms B. Evaluate 10log 12. 10log 12 = 12 10log x = x Answer: 12 Example 3

C. Evaluate log 14. log 14 ≈ 1.15 Use a calculator. Answer: Common Logarithms C. Evaluate log 14. log 14 ≈ 1.15 Use a calculator. Answer: Example 3

C. Evaluate log 14. log 14 ≈ 1.15 Use a calculator. Answer: 1.15 Common Logarithms C. Evaluate log 14. log 14 ≈ 1.15 Use a calculator. Answer: 1.15 CHECK Since 14 is between 10 and 100, log 14 is between log 10 and log 100. Since log 10 = 1 and log 100 = 2, log 14 has a value between 1 and 2. Example 3

Common Logarithms D. Evaluate log (–11). Since f (x) = logbx is only defined when x > 0, log (–11) is undefined on the set of real numbers. Answer: Example 3

Answer: no real solution Common Logarithms D. Evaluate log (–11). Since f (x) = logbx is only defined when x > 0, log (–11) is undefined on the set of real numbers. Answer: no real solution Example 3

Evaluate log 0.092. A. about 1.04 B. about –1.04 C. no real solution D. about –2.39 Example 3

Evaluate log 0.092. A. about 1.04 B. about –1.04 C. no real solution D. about –2.39 Example 3

Key Concept 4

A. Evaluate ln e4.6. ln e4.6 = 4.6 ln ex = x Answer: Natural Logarithms A. Evaluate ln e4.6. ln e4.6 = 4.6 ln ex = x Answer: Example 4

A. Evaluate ln e4.6. ln e4.6 = 4.6 ln ex = x Answer: 4.6 Natural Logarithms A. Evaluate ln e4.6. ln e4.6 = 4.6 ln ex = x Answer: 4.6 Example 4

B. Evaluate ln (–1.2). ln (–1.2) undefined Answer: Natural Logarithms Example 4

Answer: no real solution Natural Logarithms B. Evaluate ln (–1.2). ln (–1.2) undefined Answer: no real solution Example 4

C. Evaluate eln 4. eln 4 =4 elnx = x Answer: Natural Logarithms Example 4

C. Evaluate eln 4. eln 4 =4 elnx = x Answer: 4 Natural Logarithms Example 4

D. Evaluate ln 7. ln 7 ≈ 1.95 Use a calculator. Answer: Natural Logarithms D. Evaluate ln 7. ln 7 ≈ 1.95 Use a calculator. Answer: Example 4

D. Evaluate ln 7. ln 7 ≈ 1.95 Use a calculator. Answer: about 1.95 Natural Logarithms D. Evaluate ln 7. ln 7 ≈ 1.95 Use a calculator. Answer: about 1.95 Example 4

Evaluate ln e5.2. A. no real solution B. about 181.27 C. about 1.65 D. 5.2 Example 4

Evaluate ln e5.2. A. no real solution B. about 181.27 C. about 1.65 D. 5.2 Example 4

Graphs of Logarithmic Functions A. Sketch and analyze the graph of f (x) = log2 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. Construct a table of values and graph the inverse of this logarithmic function, the exponential function f –1(x) = 2x. Example 5

Graphs of Logarithmic Functions Since f (x) = log2x and f –1(x) = 2x are inverses, you can obtain the graph of f (x) by plotting the points (f –1(x), x). Example 5

Graphs of Logarithmic Functions Answer: Example 5

Graphs of Logarithmic Functions Answer: Domain: (0, ∞); Range: (–∞, ∞); x-intercept: 1; Asymptote: y-axis; Increasing: (0, ∞); End behavior: ; Example 5

Graphs of Logarithmic Functions B. Sketch and analyze the graph of Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. Construct a table of values and graph the inverse of this logarithmic function, the exponential function . Example 5

Graphs of Logarithmic Functions Since are inverses, you can obtain the graph of g (x) by plotting the points (g –1(x), x). Example 5

Graphs of Logarithmic Functions Answer: Example 5

Graphs of Logarithmic Functions Answer: Domain: (0, ∞); Range:(–∞, ∞); x-intercept: 1; Asymptote: y-axis; Decreasing: (0, ∞); End behavior: ; Example 5

Describe the end behavior of f (x) = log4 x. Example 5

Describe the end behavior of f (x) = log4 x. Example 5

Key Concept 5

Graph Transformations of Logarithmic Functions A. Use the graph of f (x) = log x to describe the transformation that results in p (x) = log (x + 1). Then sketch the graph of the function. The function is of the form p (x) = f(x + 1). Therefore, the graph p (x) is the graph of f (x) translated 1 unit to the left. Answer: Example 6

Answer: p (x) is the graph of f (x) translated 1 unit to the left. Graph Transformations of Logarithmic Functions A. Use the graph of f (x) = log x to describe the transformation that results in p (x) = log (x + 1). Then sketch the graph of the function. The function is of the form p (x) = f(x + 1). Therefore, the graph p (x) is the graph of f (x) translated 1 unit to the left. Answer: p (x) is the graph of f (x) translated 1 unit to the left. Example 6

Graph Transformations of Logarithmic Functions B. Use the graph of f (x) = log x to describe the transformation that results in m (x) = –log x – 2. Then sketch the graph of the function. The function is of the form m (x) = –f(x) – 2. Therefore, the graph of m (x) is the graph of f (x) reflected in the x-axis and then translated 2 units down. x – 2 Answer: Example 6

Graph Transformations of Logarithmic Functions B. Use the graph of f (x) = log x to describe the transformation that results in m (x) = –log x – 2. Then sketch the graph of the function. The function is of the form m (x) = –f(x) – 2. Therefore, the graph of m (x) is the graph of f (x) reflected in the x-axis and then translated 2 units down. x – 2 Answer: m (x) is the graph of f (x) reflected in the x-axis and then translated 2 units down. Example 6

Graph Transformations of Logarithmic Functions C. Use the graph of f (x) = log x to describe the transformation that results in n (x) = 5 log (x – 3). Then sketch the graph of the function. The function is of the form n (x) = 5f(x – 3). Therefore, the graph of n (x) is the graph of f (x) expanded vertically by a factor of 5 and then translated 3 units to the right. Answer: Example 6

Graph Transformations of Logarithmic Functions C. Use the graph of f (x) = log x to describe the transformation that results in n (x) = 5 log (x – 3). Then sketch the graph of the function. The function is of the form n (x) = 5f(x – 3). Therefore, the graph of n (x) is the graph of f (x) expanded vertically by a factor of 5 and then translated 3 units to the right. Answer: n (x) is the graph of f (x) expanded vertically by a factor of 5 and then translated 3 units to the right. Example 6

A. Use the graph of f (x) = ln x to describe the transformation that results in p (x) = ln (x – 2) + 1. Then sketch the graphs of the functions. A. The graph of p (x) is the graph of f (x) translated 2 units to the left and 1 unit down. B. The graph of p (x) is the graph of f (x) translated 2 units to the right and 1 unit down. C. The graph of p (x) is the graph of f (x) translated 2 units to the left and 1 unit up. D. The graph of p (x) is the graph of f (x) translated 2 units to the right and 1 unit up. Example 6

A. Use the graph of f (x) = ln x to describe the transformation that results in p (x) = ln (x – 2) + 1. Then sketch the graphs of the functions. A. The graph of p (x) is the graph of f (x) translated 2 units to the left and 1 unit down. B. The graph of p (x) is the graph of f (x) translated 2 units to the right and 1 unit down. C. The graph of p (x) is the graph of f (x) translated 2 units to the left and 1 unit up. D. The graph of p (x) is the graph of f (x) translated 2 units to the right and 1 unit up. Example 6

Use Logarithmic Functions A. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R , where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. Find the intensity of an earthquake with an amplitude of 250 microns, a period of 2.1 seconds, and B = 5.4. Example 7

The intensity of the earthquake is about 7.5. Use Logarithmic Functions R = Original Equation = a = 250, T = 2.1, and B = 5.4 ≈ 7.5 The intensity of the earthquake is about 7.5. Answer: Example 7

The intensity of the earthquake is about 7.5. Use Logarithmic Functions R = Original Equation = a = 250, T = 2.1, and B = 5.4 ≈ 7.5 The intensity of the earthquake is about 7.5. Answer: about 7.5 Example 7

Use Logarithmic Functions B. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R , where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. A city is not concerned about earthquakes with an intensity of less than 3.5. An earthquake occurs with an amplitude of 125 microns, a period of 0.33 seconds, and B = 1.2. What is the intensity of the earthquake? Should this earthquake be a concern for the city? Example 7

R = Original Equation = a = 125, T = 0.33, and B = 1.2 ≈ 3.78 Use Logarithmic Functions R = Original Equation = a = 125, T = 0.33, and B = 1.2 ≈ 3.78 The intensity of the earthquake is about 3.78. Since 3.78 ≥ 3.5, the city should be concerned. Answer: Example 7

R = Original Equation = a = 125, T = 0.33, and B = 1.2 ≈ 3.78 Use Logarithmic Functions R = Original Equation = a = 125, T = 0.33, and B = 1.2 ≈ 3.78 The intensity of the earthquake is about 3.78. Since 3.78 ≥ 3.5, the city should be concerned. Answer: about 3.78 Example 7

Use Logarithmic Functions C. EARTHQUAKES The Richter scale measures the intensity R of an earthquake. The Richter scale uses the formula R , where a is the amplitude (in microns) of the vertical ground motion, T is the period of the seismic wave in seconds, and B is a factor that accounts for the weakening of seismic waves. Earthquakes with an intensity of 6.1 or greater can cause considerable damage to those living within 100 km of the earthquake’s center. Determine the amplitude of an earthquake whose intensity is 6.1 with a period of 3.5 seconds and B = 3.7. Example 7

Use Logarithmic Functions Use a graphing calculator to graph and R = 6.1 on the same screen and find the point of intersection. Example 7

Use Logarithmic Functions An earthquake with an intensity of 6.1, a period of 3.5, and a B-value of 3.5 has an amplitude of about 879 microns. Answer: Example 7

Use Logarithmic Functions An earthquake with an intensity of 6.1, a period of 3.5, and a B-value of 3.5 has an amplitude of about 879 microns. Answer: about 879 microns Example 7

SOUND The intensity level of a sound, measured in decibels, can also be modeled by the equation d (w) = 10 log (1012w) where w is the intensity of the sound in watts per square meter. If the intensity of the sound of a chain saw is 0.1 watts per square meter, what is the intensity level of the sound in decibels? A. 110 decibels B. 100 decibels C. 90 decibels D. 80 decibels Example 7

SOUND The intensity level of a sound, measured in decibels, can also be modeled by the equation d (w) = 10 log (1012w) where w is the intensity of the sound in watts per square meter. If the intensity of the sound of a chain saw is 0.1 watts per square meter, what is the intensity level of the sound in decibels? A. 110 decibels B. 100 decibels C. 90 decibels D. 80 decibels Example 7

End of the Lesson