Section 4.5 Logarithmic and Exponential Equations
The logarithm and exponential functions are onetoone functions and are inverse functions of each
other. Hence
•
If
a
x
=
a
y
, then
x
=
y
.
•
If log
a
M
= log
a
N
, then
M
=
N
.
•
a
log
a
x
=
x
•
log
a
a
x
=
x
The following examples illustrate the preceding rules.
Example.
Solve log
3
(3
x

2) = 2.
Solution. log
3
(3
x

2) = 2
3
log
3
(3
x

2)
= 3
2
= 9
3
x

2 = 9
x
=
11
3
Example.
Solve 3 log
2
x
=

log
2
27.
Solution. 3 log
2
x
=

log
2
27
log
2
x
3
= log
2
27

1
x
3
= 27

1
=
1
27
x
=
1
3
Example.
Solve 5
1

2
x
=
1
5
.
Solution. 5
1

2
x
=
1
5
= 5

1
implies
1

2
x
=

1
x
= 1
Example.
Solve 2
x
+1
= 5
1

2
x
.
Solution. 2
x
+1
= 5
1

2
x
ln 2
x
+1
= ln 5
1

2
x
(
x
+ 1) ln 2 = (1

2
x
) ln 5
x
(ln 2 + 2 ln 5) = ln 5

ln 2
1
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x
(ln 50) = ln
5
2
x
=
ln
5
2
ln 50
≈
0
.
230 (Calculator)
Example.
Solve
e
x
+3
=
π
x
.
Solution.
e
x
+3
=
π
x
ln
e
x
+3
= ln
π
x
(
x
+ 3) ln
e
=
x
ln
π
(ln
e
= 1)
x
+ 3 =
x
ln
π
x
(ln
π

1) = 3
x
=
3
ln
π

1
≈
20
.
728
(Calculator)
Example.
Solve log
2
8
x
= 3.
Solution. log
2
8
x
= 3
log
2
(2
3
)
x
= 3
log
2
2
3
x
= 3
3
x
= 3
x
= 1
Example.
Solve log
2
(3
x
+ 2)

log
4
x
= 3.
Solution. log
2
(3
x
+ 2)

log
4
x
= 3
log
2
(3
x
+ 2)

log
2
x
log
2
4
= 3
(Change of base formula)
2
log
2
(3
x
+2)

log
2
x
2
= 2
3
= 8
(log
2
4 = 2)
2
log
2
(3
x
+2)
·
2

log
2
√
x
= 8
(2
a

b
= 2
a
2

b
)
(3
x
+ 2)
·
1
√
x
= 8
3
x
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 Fall '11
 Kutter
 Algebra, Exponential Function, Equations, Exponential Functions, Inverse Functions, Natural logarithm, Logarithm, Example. Solve

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