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115.3 Assignment#3 Solutions-1
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115.3 Assignment#3 Solutions
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ln x 2 − 1
loge x 2 − 1
=
x −1 =
loge 2
ln 2
(a) log2 x − 1
Solution:
log2
(b) log3 (5x + 1)
Solution:
log3 (5x + 1) =
2
2
loge (5x + 1)
ln(5x + 1)
=
loge 3
ln 3
log (x + 2)
ln(x + 2)
e
=
log10 x 2 − 1 =
loge 10
ln 10
2
log 2x 2 − 1
ln
2x
−
1
e
=
(d) log2 2x 2 − 1 Solution: log2 2x 2 − 1 =
loge 2
ln 2
(c) log(x + 2)
Solution:
y
1.3-2 Sketch the graph of y = −(x − 2) + 1 without using a graphing calculator.
2
Solution:
Starting with the graph of y = x 2 in black,
we get the graph shifted 2 units to the right in blue: y = (x − 2)2 ,
that graph reflected about the x-axis in green: y = −(x − 2)2 ,
and finally that graph shifted up 1 unit in red: y = −(x − 2)2 + 1.
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1.2-86 Write the following expressions in terms of base e.
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5
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3
2
1
0
-1 1 2 3 4
-4-3-2-10
-2
-3
-4
-5
-6
-7
-8
-9
-10
x
y
Solution:
Starting with the graph of y = ex in black,
we get that reflected about the y-axis graph in red: y = e−x .
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1.3-12 Sketch the graph of y = ex without using a graphing calculator.
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-4-3-2-10 1 2 3 4
115.3 Assignment#3 Solutions-2
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x
1.3-34 Find the following numbers on a number line that is on a logarithmic scale (base 10):
0.03,0.7,1,2,5,10,1.7,100,150, and 2000.
Solution:
0.03
1
-2
10
2
3 4 5 6 7 891
-1
10
2
0.7 1
2
3 4 5 6 7 891
2
0
10
5
10
3 4 5 6 7 891
1
10
17
2
100150
3 4 5 6 7 891
2
10
2000
2
1
3 4 5 6 7 89
3
10
2
3 4 5 6 7 89
4
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115.3 Assignment#3 Solutions-3
1.3-36 Find the following numbers on a number line that is on a logarithmic scale (base 10):
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(a) 10−3 , 2 × 10−3 , 5 × 10−3
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(b) 10−1 , 2 × 10−1 , 5 × 10−1
(c) 102 , 2 × 102 , 2 × 102
(d) Using (a)-(c), how many units (on a logarithmic scale) is 2 × 10−3 from 10−3 (2 × 10−2 from 10−2 , 2 × 102
from 102 )?
(e) Using (a)-(c), how many units (on a logarithmic scale) is 5 × 10−3 from 10−3 (5 × 10−2 from 10−2 , 5 × 102
from 102 )?
Solution:
(d) 1 , (e) 4
0.03
1
-2
10
2
3 4 5 6 7 891
-1
10
2
0.7 1
2
3 4 5 6 7 891
2
0
10
5
10
3 4 5 6 7 891
1
10
17
2
100150
3 4 5 6 7 891
2
10
2000
2
1
3 4 5 6 7 89
3
10
1.3-38 Both the La Plata river dolphin and the sperm whale are marine mammals having teeth. A La Plata river
dolphin weighs between 30 and 50 kg, whereas a sperm whale weighs between 35,000 and 40,000 kg. How
many orders of magnitude greater is the weight of the sperm whale?
40, 000
=3
Solution: log10
40
2
3 4 5 6 7 89
4
10
115.3 Assignment#3 Solutions-4
(x1 , y1 ) = (−1, 4),
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1.3-44 When log y is graphed as a function of x on log-linear paper, a straight line results. Graph straight lines,
each given by two points, on a log-linear plot, and determine the functional relationship.
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(x2 , y2 ) = (2, 8)
Since the graph is a straight line on log-linear graph paper, we have Y=mx+b for constants
m and b which we must find.
At (x1 , y1 ) = (−1, 4) we have log 4 = m(−1) + b, and
at (x2 , y2 ) = (2, 8) we have log 8 = m(2) + b, so on subtracting the first equation from the second, we get
log 84
log 8 − log 4
log 2
log 8 − log 4 = 3m or m =
=
=
0.100
3
3
3
and we have the equation Y =
log 2
7
3 x+ 3
= 2 log 2 +
log 2
3
=
7
3
log 2 0.702
log 2
7
x + log 2.
3
3
Exponentiating, we get
y = 10Y = 10log y = 10
log 2
3
log 2
log 2 x 1 x
7 x
7
7
3
3
= 10 3 log 2 × 10 3
= 10log 2 × 10log 2
= 23 × 23
5.04 × 1.26x
Y=log y
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-2
-1
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2
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Solution:
Next, we have b = log 4 + m = log 4 +
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115.3 Assignment#3 Solutions-5
y = 4 × 105x .
Solution:
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1.3-48 Use a logarithmic transformation to find a linear relationship between the given quantities and graph the
resulting linear relationship on a log-linear plot:
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We have Y = log(4 × 10
) = log 4 + 5x
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Y=log y
5x
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115.3 Assignment#3 Solutions-6
Solution:
We have Y = log y = log 5, so y = 5 .
y
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1
1
2
3
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7 8 9 10
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1.3-56 When log y is graphed as a function of log x a straight line results. Graph straight lines, each given by two
points on a log-log plot, and determine the functional relationship. (The original x-y coordinates are given.)
(x1 , y1 ) = (3, 5), (x2 , y2 ) = (1, 5)
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115.3 Assignment#3 Solutions-7
y = 3x 2
Solution:
Y = log y = log(3x 2 ) = log 3 + 2 log x = log 3 + 2X, where X = log x.
y
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1.3-60 Use a logarithmic transformation to find a linear relationship between the given quantities and graph the
resulting linear relationships on a log-log plot.
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115.3 Assignment#3 Solutions-8
g(s) = 1.8e−0.2s
Solution:
G = log g(s) = log 1.8e−0.2s = log 1.8 + log e−0.2s = log 1.8 − 0.2s log e = log 1.8 − (0.2 log e)s which we
plot on log-linear paper:
G=log g
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1
-1
0
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s
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1.3-68 Use a logarithmic transformation to find a linear relationship between the given quantities and determine
whether a log-log or a log-linear plot should be used to graph the resulting relationship
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115.3 Assignment#3 Solutions-9
Solution:
L(c) = 1.7 × 102.3c
log L(c) = log 1.7 × 102.3c = log 1.7 + log 102.3c = log 1.7 + 2.3c which we plot on
log-linear paper:
log L
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1
-1
0
1
c
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1.3-72 Use a logarithmic transformation to find a linear relationship between the given quantities and determine
whether a log-log or a log-linear plot should be used to graph the resulting relationship
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