Over Lesson 3-1
A. Sketch the graph of f(x) = 3x + 1.
A.
C.
B.
D.
Over Lesson 3-1
A. Sketch the graph of f(x) = 3x + 1.
A.
C.
B.
D.
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
Answer: 4
Equality property of exponents.
Evaluate Logarithms
B. Evaluate
=y
5y =
5y = 5–3
y = –3
Answer: –3
.
Let
= y.
Write in exponential form.
= 5–3
Equality property of exponents.
Evaluate Logarithms
C. Evaluate
=y
3y =
3y = 3–3
y = –3
Answer: –3
.
Let log3 = y.
Write in exponential form.
= 3–3
Equality property of exponents.
Evaluate Logarithms
D. Evaluate log17 17.
log1717 = y
Let log1717 = y.
17y = 17
Write in exponential form.
17y = 171
17= 171
y = 17
Answer: 1
Equality property of exponents.
Evaluate
A. –4
B. 4
C. –2
D. 2
.
Evaluate
A. –4
B. 4
C. –2
D. 2
.
Apply Properties of Logarithms
A. Evaluate log8 512.
log8512 = log883
=3
Answer: 3
83 = 512
logbbx = x
Apply Properties of Logarithms
B. Evaluate 22log22 15.2.
22log22 15.2 =15.2
Answer: 15.2
blogbx = x
Evaluate 7log7 4.
A. 4
B. 7
C. 47
D. 74
Evaluate 7log7 4.
A. 4
B. 7
C. 47
D. 74
Common Logarithms
A. Evaluate log 10,000.
log10,000 = log104
=4
Answer: 4
10,000 = 104
log10x = x
Common Logarithms
B. Evaluate 10log 12.
10log 12 = 12
Answer: 12
10log x = x
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.
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
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
Answer: 4.6
ln ex = x
Natural Logarithms
B. Evaluate ln (–1.2).
ln (–1.2) undefined
Answer: no real solution
Natural Logarithms
C. Evaluate eln 4.
eln 4 =4
Answer: 4
elnx = x
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
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.
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).
Graphs of Logarithmic Functions
Answer: Domain: (0, ∞); Range: (–∞, ∞);
x-intercept: 1; Asymptote: y-axis;
Increasing: (0, ∞);
End behavior:
;
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
.
Graphs of Logarithmic Functions
Since
are inverses, you
can obtain the graph of g(x) by plotting the points
(g –1(x), x).
Graphs of Logarithmic Functions
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.
Describe the end behavior of f(x) = log4 x.
A.
B.
C.
D.
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.
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.
Answer: m(x) is the
graph of f(x)
reflected in the
x-axis and then
translated 2
units down.
x–2
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.
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.
C.
The graph of p (x) is the graph
of f (x) translated 2 units to the
left and 1 unit up.
B.
The graph of p (x) is the graph
of f (x) translated 2 units to
the right and 1 unit down.
D.
The graph of p (x) is the graph
of f (x) translated 2 units to the
right and 1 unit up.
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.
C.
The graph of p (x) is the graph
of f (x) translated 2 units to the
left and 1 unit up.
B.
The graph of p (x) is the graph
of f (x) translated 2 units to
the right and 1 unit down.
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.
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
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?
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
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.
Use Logarithmic Functions
Use a graphing calculator to graph
and R = 6.1 on the same screen and find the point of
intersection.
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
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
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
• logarithmic function with base b
• logarithm
• common logarithm
• natural logarithm