Review Homework (3.1 Graphing exponential functions)

You graphed and analyzed exponential functions. (Lesson 3-1)

• Evaluate expressions involving logarithms.

• Sketch and analyze graphs of logarithmic functions.

• logarithmic function with base b

• logarithm

• common logarithm

• natural logarithm

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

Evaluate Logarithms

B. Evaluate .

5y = Write in exponential form.

= y Let = y.

5y = 5–3 = 5–3

y = –3 Equality property of exponents.

Answer: –3

Evaluate Logarithms

C. Evaluate .

= y Let log3 = y.

y = –3 Equality property of exponents.

3y= 3–3 = 3–3

3y = Write in exponential form.

Answer: –3

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

A. –4

B. 4

C. –2

D. 2

Evaluate .

Apply Properties of Logarithms

A. Evaluate log8 512.

log8512= log883 83 =

512

=3 logbbx = x

Answer: 3

Apply Properties of Logarithms

B. Evaluate 22log22 15.2.

22log22 15.2 =15.2 blogbx = x

Answer: 15.2

Evaluate 7log7 4.

A. 4

B. 7

C. 47

D. 74

Common Logarithms

A. Evaluate log 10,000.

log10,000= log104

10,000 = 104

= 4 log10x = x

Answer: 4

Common Logarithms

B. Evaluate 10log 12.

10log 12 = 12 10log x = x

Answer: 12

Common Logarithms

C. Evaluate log 14.

log 14 ≈ 1.15 Use a calculator.

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.

Answer: 1.15

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

Evaluate log 0.092.

A. about 1.04

B. about –1.04

C. no real solution

D. about –2.39

Natural Logarithms

A. Evaluate ln e4.6.

ln e4.6 = 4.6 ln ex = x

Answer: 4.6

Natural Logarithms

B. Evaluate ln (–1.2).

ln (–1.2) undefined

Answer: no real solution

Natural Logarithms

C. Evaluate eln 4.

eln 4 =4 elnx = x

Answer: 4

Natural Logarithms

D. Evaluate ln 7.

ln 7 ≈ 1.95 Use a calculator.

Answer: about 1.95

Evaluate ln e5.2.

A. no real solution

B. about 181.27

C. about 1.65

D. 5.2

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.


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).


Answer: Domain: (0, ∞); Range: (–∞, ∞);

x-intercept: 1; Asymptote: y-axis;

Increasing: (0, ∞);

End behavior: ;


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 .


Since are inverses, you

can obtain the graph of g (x) by plotting the points

(g –1(x), x).


Answer: Domain: (0, ∞); Range:(–∞, ∞);

x-intercept: 1; Asymptote: y-axis;

Decreasing: (0, ∞);

End behavior:

;

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

A.

B.

C.

D.

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.

Graph Transformations of Logarithmic Functions

Answer: p (x) is the graph of f (x) translated 1 unit to the left.

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.


Answer: m (x) is the graph of f (x) reflected in the x-axis and then translated 2 units down.

x – 2

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.

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.

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.


Answer: about 7.5

≈ 7.5

R = Original Equation

= a = 250, T = 2.1, and B = 5.4

The intensity of the earthquake is about 7.5.


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?


Answer: about 3.78

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.


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.


Use a graphing calculator to graph

and R = 6.1 on the same screen and find the point of

intersection.


Answer: about 879 microns

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.

Review Homework (3.1 Graphing exponential functions)

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