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236 Chapter 3 Exponential and Logarithmic Functions
122.
y 1 1.05 123.
y 2 lnx 2 1 2 lnx 2
2x 2 e 2x 2xe 2x 0
2x 2 2xe 2x 0
2
2x 2 2x 0
since e 2x 0
y 1
y 2
0
0
12
2xx 1 0
x 0, 1
The graphs intersect at 10.166, 1.05.
124.
x 2 e x 2xe x 0 125.
x 2 2xe x 0
xe x e x 0
x 1e x 0
x 2 2x 0
since e x 0
x 1 0
since e x 0
xx 2 0
x 1
x 0, 2
126.
e 2x 2xe 2x 0 127.
2x ln x x 0
1 2xe 2x 0
x2 ln x 1 0
1 2x 0
since e 2x 0
2 ln x 1 0
since x > 0
x 1 2
ln x 1 2
x e 12 0.607
128.
1 ln x
x 2 0 129.
1 ln x
2
0
1 ln x 0 x > 0
ln x 1
x e 2.718
1 ln x 0
ln x 1
x e 1 1 e 0.368
130.
2x ln
1
x x 0
x 2 ln
1
x 1 0
2 ln
1
1 0 since x > 0
x
ln
1
x 1 2
1
x e12
x e 12 0.607
131. (a)
(b)
2000 1000e 0.075t
2 e 0.075t
ln 2 0.075t
t ln 2 9.24 years
0.075
3000 1000e 0.075t
3 e 0.075t
ln 3 0.075t
t ln 3 14.65 years
0.075
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