Some 1.7 solutions
September 17, 2006
23

26: Solve the following equations for x. Plug in your answer to check.
Key idea  You want to isolate your exponential function and then take
natural log of both sides and then hopefully the equation will be easier to
solve. You use the log to get the variable out of the exponent. The main
identity involved is
ln
(
a
r
) =
r
*
ln
(
a
) which is the one that gets the vari
able out of the exponent.
Sometimes you may want to use the identity
ln
(
ab
) =
ln
(
a
) +
ln
(
b
), though using that one can often be avoided.
23: 7
e
3
x
= 21
e
3
x
= 3 (divide both sides by 3 to isolate the exponential expression)
ln
(
e
3
x
) =
ln
(3) (take natural log of both sides)
3
x
≈
1
.
096 (use the identity and plug
ln
(3) into your calculator)
x
≈
0
.
366
24: 4
e
2
x
+1
= 20
e
2
x
+1
= 5 (divide both sides by 4)
ln
(
e
2
x
+1
) =
ln
(5) (take natural log of both sides)
2
x
+ 1
≈
1
.
609 (use the identity and put
ln
(5) into your calculator)
2
x
≈
0
.
609 (subtract 1 from both sides)
x
≈
0
.
305 (divide both sides by 2)
25: 4
e

2
x
+1
= 7
e
3
x
e

2
x
+1
=
7
4
e
3
x
(divide both sides by 4)
ln
(
e

2
x
+1
) =
ln
(
7
4
e
3
x
) (take natural log of both sides)

2
x
+ 1 =
ln
(
7
4
) +
ln
(
e
3
x
) (use identities)

2
x
+ 1
≈
0
.
6 + 3
x
(more identities and some calculator usage)
1
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5
x
≈ 
0
.
4 (subtract 3
x
from both sides and subtract 1 from both sides)
x
≈
0
.
08 (divide by

5).
If you are determined not to use the identity
ln
(
ab
) =
ln
(
a
)+
ln
(
b
), you can
use an property of exponents as follows:
4
e

2
x
+1
= 7
e
3
x
4 =
7
e
3
x
e

2
x
+1
(divide both sides by
e

2
x
+1
)
4 = 7
e
3
x

(

2
x
+1)
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 Spring '09
 levitt
 Math, Exponential Function, Equations, 100,000 years, α α, 000 years, 5680 years

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