06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 31
Home
Quit
Lesson 5.6 Exercises, pages 405–410
A
3. Approximate the value of each logarithm, to the nearest thousandth.
a) log29
b) log2100
Use the change of base formula to change the base of the logarithms to
base 10.
log29 ⴝ
log 9
log2100 ⴝ
log 2
log 100
log 2
ⴝ 6.6438. . .
⬟ 6.644
ⴝ 3.1699. . .
⬟ 3.170
4. Order these logarithms from greatest to least:
log280, log3900, log45000, log510 000
Write each logarithm to base 10, then calculate its value.
log280
ⴝ
log 80
log 2
ⴝ 6.3219. . .
log3900
ⴝ
log 900
log 3
ⴝ 6.1918. . .
log45000
ⴝ
log 5000
log 4
ⴝ 6.1438. . .
log510 000
ⴝ
log 10 000
log 5
ⴝ 5.7227. . .
From greatest to least: log280, log3900, log45000, log510 000
©P DO NOT COPY.
5.6 Analyzing Logarithmic Functions—Solutions
31
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 32
B
5. Approximate the value of each logarithm, to the nearest thousandth.
Write the related exponential expression.
b) log3 a2b
1
a) log7400
log7400 ⴝ
log 400
log3 a b ⴝ
1
2
log 7
ⴝ 3.0790. . .
⬟ 3.079
So, 400 ⬟ 73.079
log 0.5
log 3
ⴝ ⴚ0.6309. . .
⬟ ⴚ0.631
So,
1
⬟ 3ⴚ0.631
2
6. a) Use technology to graph y ⫽ log5x. Sketch the graph.
Graph: y ⴝ
log x
log 5
b) Identify the intercepts and the equation of the asymptote of the
graph, and the domain and range of the function.
From the graph, the x-intercept is 1. There is no y-intercept. The equation
of the asymptote is x ⴝ 0. The domain of the function is x>0. The range
of the function is y ç ⺢.
c) Choose the coordinates of two points on the graph. Multiply
their x-coordinates and add their y-coordinates. What do you
notice about the new coordinates? Explain the result.
From the TABLE, two points on the graph have coordinates:
(5, 1) and (25, 2)
The product of the x-coordinates is 125. The sum of the y-coordinates is 3.
The new coordinates are (125, 3), which is also a point on the graph.
The logarithm of the product of two numbers is the sum of the logarithms
of the numbers.
7. a) Use a graphing calculator to graph y ⫽ log2x, y ⫽ log4x, and
y ⫽ log8x. Sketch the graphs.
Graph: y ⴝ
32
log x
log 2
,y ⴝ
log x
log 4
, and y ⴝ
log x
log 8
5.6 Analyzing Logarithmic Functions—Solutions
DO NOT COPY. ©P
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 33
b) In part a, what happened to the graph of y ⫽ logbx, b > 0,
b Z 1, as the base changed?
As b increases, from b ⴝ 2, the graph of y ⴝ logbx is compressed
log x
vertically by a factor of:
log b
log x
ⴝ
log 2
log b
.
log 2
y g(x)
8. a) The graphs of a logarithmic function and its transformation
4
image are shown. The functions are related by translations, and
corresponding points are indicated. Identify the translations.
A'
y
B'
B
2
y f(x)
A
From A to A’, the translations are
3 units left and 1 unit up.
The same translations relate B and B’.
2
0
x
2
4
6
8
2
b) Given that f(x) ⫽ log2x, what is g(x)? Justify your answer.
After translations, the image of the graph of y ⴝ log2x has equation:
Substitute: k ⴝ 1 and h ⴝ ⴚ3
y ⴚ k ⴝ log2(x ⴚ h)
The image graph has equation y ⴚ 1 ⴝ log2(x ⴙ 3); or
y ⴝ log2(x ⴙ 3) ⴙ 1
So, g(x) ⴝ log2(x ⴙ 3) ⴙ 1
9. a) How is the graph of y ⫽ 2 log2(2x ⫺ 8) related to the graph of
y ⫽ log2x? Sketch both graphs on the same grid.
Compare y ⴝ 2 log22(x ⴚ 4) with y ⴚ k ⴝ c log2 d(x ⴚ h):
k ⴝ 0, c ⴝ 2, d ⴝ 2, and h ⴝ 4
Write y ⴝ 2 log2(2x ⴚ 8) as y ⴝ 2 log22(x ⴚ 4). The graph of this
function is the image of the graph of y ⴝ log2x after a vertical stretch
1
by a factor of 2, a horizontal compression by a factor of , then a
2
translation of 4 units right.
y 2 log2(2x 8)
y
4
y log2x
2
x
0
x
Use the general transformation: (x, y) corresponds to a ⴙ h, cy ⴙ kb
d
2
x
The point (x, y) on y ⴝ log2x corresponds to the point a ⴙ 4, 2yb on
2
4
2
6
8
y ⴝ 2 log22(x ⴚ 4).
(x, y)
x
a ⴙ 4, 2yb
2
(0.5, ⴚ1)
(4.25, ⴚ2)
(1, 0)
(4.5, 0)
(2, 1)
(5, 2)
(4, 2)
(6, 4)
(8, 3)
(8, 6)
©P DO NOT COPY.
5.6 Analyzing Logarithmic Functions—Solutions
33
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 34
b) Identify the intercepts and the equation of the asymptote of the
graph of y ⫽ 2 log2(2x ⫺ 8), and the domain and range of the
function. Use graphing technology to verify.
From the graph, there is no y-intercept.
From the table, the x-intercept is 4.5.
The equation of the asymptote is x ⴝ 4.
The domain of the function is x>4.
The range of the function is y ç ⺢.
1
1
10. a) Graph y = - 4 log2 a2xb + 1.
Compare y ⴚ 1 ⴝ ⴚ log2 a xb
1
4
1
2
with y ⴚ k ⴝ c log2 d(xⴚh):
1
4
1
2
k ⴝ 1, c ⴝ ⴚ , d ⴝ , and h ⴝ 0
2
0
y
1
1
y log2 x 1
4
2
x
4
6
8
( )
2
2
Use the general transformation:
(x, y) corresponds to a ⴙ h, cy ⴙ kb
x
d
The point (x, y) on y ⴝ log2x corresponds to the point a2x, ⴚ y ⴙ 1b
on y ⴝ ⴚ log2 a xb ⴙ 1.
1
4
1
2
(x, y)
1
a2x, ⴚ y ⴙ 1b
4
(0.25, ⴚ2)
(0.5, 1.5)
(0.5, ⴚ1)
(1, 1.25)
(1, 0)
(2, 1)
(2, 1)
(4, 0.75)
(4, 2)
(8, 0.5)
1
4
b) Identify the intercepts and the equation of the asymptote of the
graph of y = - 4 log2 a2xb + 1, and the domain and range of
the function.
1
1
From the graph, there is no y-intercept.
Use the TABLE feature on a graphing calculator; the x-intercept is 32.
The equation of the asymptote is x ⴝ 0.
The domain of the function is x>0.
The range of the function is y ç ⺢.
34
5.6 Analyzing Logarithmic Functions—Solutions
DO NOT COPY. ©P
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 35
11. Graph each function below, then identify the intercepts and the
equation of the asymptote of the graph, and the domain and range
of the function.
a) y ⫽ ⫺log2(x ⫹ 4) ⫺ 3
Compare y ⴙ 3 ⴝ ⴚlog2(x ⴙ 4) with y ⴚ k ⴝ c log2 d(x ⴚ h):
k ⴝ ⴚ3, c ⴝ ⴚ1, d ⴝ 1, and h ⴝ ⴚ4
x
Use the general transformation: (x, y) corresponds to a ⴙ h, cy ⴙ kb
d
The point (x, y) on y ⴝ log2x corresponds to the
point (x ⴚ 4, ⴚy ⴚ 3) on y ⴝ ⴚlog2(x ⴙ 4) ⴚ 3.
(x, y)
(x ⴚ 4, ⴚy ⴚ 3)
(0.25, ⴚ2)
(ⴚ3.75, ⴚ1)
(0.5, ⴚ1)
(ⴚ3.5, ⴚ2)
(1, 0)
(ⴚ3, ⴚ3)
(2, 1)
(ⴚ2, ⴚ4)
(4, 2)
(0, ⴚ5)
(8, 3)
(4, ⴚ6)
2
y
x
2
0
2
4
2
y log2(x 4)3
4
6
From the graph, the x-intercept is approximately ⴚ3.9.
From the table, the y-intercept is ⴚ5.
The equation of the asymptote is x ⴝ ⴚ4.
The domain of the function is x>ⴚ4.
The range of the function is y ç ⺢.
b) y ⫽ 4 log2(⫺x ⫺ 3)
Write y ⴝ 4 log2(ⴚx ⴚ 3) as y ⴝ 4 log2[ⴚ(x ⴙ 3)].
Compare y ⴝ 4 log2[ⴚ(x ⴙ 3)] with y ⴚ k ⴝ c log2 d(x ⴚ h):
k ⴝ 0, c ⴝ 4, d ⴝ ⴚ1, and h ⴝ ⴚ3
x
Use the general transformation: (x, y) corresponds to a ⴙ h, cy ⴙ kb
d
The point (x, y) on y ⴝ log2x corresponds to the point (ⴚx ⴚ 3, 4y) on
y ⴝ 4 log2(ⴚx ⴚ 3).
(x, y)
(ⴚx ⴚ 3, 4y)
(0.25, ⴚ2)
(ⴚ3.25, ⴚ8)
(0.5, ⴚ1)
(ⴚ3.5, ⴚ4)
(1, 0)
(ⴚ4, 0)
(2, 1)
(ⴚ5, 4)
(4, 2)
(ⴚ7, 8)
8
y
4
x
6
4
y 4 log2(x 3)
2
0
4
8
From the table, the x-intercept is ⴚ4.
From the graph, there is no y-intercept.
The equation of the asymptote is x ⴝ ⴚ3.
The domain of the function is x<ⴚ3.
The range of the function is y ç ⺢.
©P DO NOT COPY.
5.6 Analyzing Logarithmic Functions—Solutions
35
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 36
C
1
12. Graph the function y ⫽ ⫺3 log3(⫺2x ⫺ 4) ⫹ 5, then identify the
intercepts, the equation of the asymptote, and the domain and
range of the function.
1
3
1
3
Write y ⴝ ⴚ log3(ⴚ2x ⴚ 4) ⴙ 5 as y ⴚ 5 ⴝ ⴚ log3[ⴚ2(x ⴙ 2)].
1
3
y Compare y ⴚ 5 ⴝ ⴚ log3[ⴚ2(x ⴙ 2)] with y ⴚ k ⴝ c log3d(x ⴚ h):
1
log (2x 4) 5
3
4
x
Use the general transformation: (x, y) corresponds to a ⴙ h, cy ⴙ kb
d
2
The point (x, y) on y ⴝ log3x corresponds to the point
x
6
1
1
1
aⴚ x ⴚ 2, ⴚ y ⴙ 5b on y ⴝ ⴚ log3(ⴚ2x ⴚ 4) ⴙ 5.
2
3
3
1
1
aⴚ x ⴚ 2, ⴚ y ⴙ 5b
2
3
1
a , ⴚ2b
9
aⴚ
1
a , ⴚ1b
3
13 16
aⴚ , b
6 3
(1, 0)
5
aⴚ , 5b
2
(3, 1)
7 14
aⴚ , b
2 3
(9, 2)
aⴚ
37 17
, b
18 3
y
6
1
k ⴝ 5, c ⴝ ⴚ , d ⴝ ⴚ2, and h ⴝ ⴚ2
3
(x, y)
8
4
2
0
To determine the x-intercept, solve the equation:
1
3
1
ⴚ5 ⴝ ⴚ log3(ⴚ2x ⴚ 4)
3
0 ⴝ ⴚ log3(ⴚ2x ⴚ 4) ⴙ 5
15 ⴝ
ⴚ2x ⴚ 4 ⴝ
ⴚ2x ⴝ
xⴝ
log3(ⴚ2x ⴚ 4)
315
14 348 911
ⴚ7 174 455.5
Write in exponential form.
13 13
, b
2 3
From the graph, there is no y-intercept.
The equation of the asymptote is x ⴝ ⴚ2.
The domain of the function is x<ⴚ2.
The range of the function is y ç ⺢.
36
5.6 Analyzing Logarithmic Functions—Solutions
DO NOT COPY. ©P