3.3 equations involving expo log
18
Chapter 3.3 Equations Involving Exponential and Logarithmic Expressions 1
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Transcript of 3.3 equations involving expo log
Chapter 3.3 Equations Involving Exponential
and Logarithmic Expressions
1
Laws of Logarithm
For all , 0, , 1,
, 0, .
1. log log log
2. log log log
3. log log
log4. log
log
b b b
b b b
rb b
mb
m
b m b m
p q r
pq p q
pp q
q
p r p
pp
b
2
Example 3.3.1
2 2 2
2 2
2
2
2
Express the following as single logarithm.
1. log 1 log 3 log 3
log 1 3 log 3
1 3log
3
3 3log
3
a a a
a a a
a a
a
a a
a
3
32 32
32
3
2
32
2 3
32. ln 2ln 3ln
2
ln ln ln
ln ln
ln
x y z
x y z
xz
y
x
y z
4
Example 3.3.2
2 2
2 2
Suppose log 2
log 3
log 5
Express the following in terms of , , and .
1. log 180
log 2 3 5
log 2 log 3 log 5
2log 2 2log 3 log 5
2 2
b
b
b
b
b
b b b
b b b
m
n
o
m n o
m n o
5
2
2
log 2 log 3 log 5
2. log 0.75
75log
100
3log
2
log 3 log 2
log 3 2log 2
2
b b b
b
b
b
b b
b b
m n o
n m
6
12
12 2
2
2
log 2 log 3 log 5
3. log 60
log 60
log 2 3 5
1log 2 3 5
21
log 2 log 3 log 521
2log 2 log 3 log 521
22
b b b
b
b
b
b
b b b
b b b
m n o
m n o
7
Solving Equations
To solve equations involving exponential
and logarithmic expressions use:
1. the laws of logarithm
2. logxby b x y
10
Example 3.3.3
4 1
4 13 2
12 2 2
Solve the following equations.
1. 27 9
3 3
3 3
12 2 2
10 2
1 1
5 5
x x
x x
x x
x x
x
x SS
11
1 2
1 23 2
3 3 4 2
3 3 4 2
2. 8 25
2 5
2 5
log 2 log 5
3 3 log2 4 2 log 5
3 log2 3log2 4log 5 2 log 5
3 log2 2 log 5 4log 5 3log2
3log2 2log 5 4log 5 3log2
x x
x x
x x
x x
x x
x x
x x
x
12
3log2 2log 5 4log 5 3log2
4log 5 3log2
3log2 2log 5
4log 5 3log2
3log2 2log 5
x
x
SS
13
23
2
2
3
2
3 43. 3 4 1
3 2
3 4 3 3 2 3 4 3 1
3 4 3 2 3 log 4 no sol.
3 3 3 4 0
3 3 3 4 0 log 4
let 3
3 4 0
4 1 0
xx
x
x x x x x
x x x
x x
x x
x
u u
x
SS
u
u u
u u
logxby b x y
14
3
2
4. log 5 4 2
5 4 3
5 4 9
5 5
1
1
x
x
x
x
x
SS
log xbx y y b
15
9 9
9
29
32 2
2
2
35. log log 6
23
log 623
log 62
6 9
6 27
6 27 0
9 3 0
x x
x x
x x
x x
x x
x x
x x
log xbx y y b
16
9 9
9
35. log log 6
2
9 3 0
9 3
3 is an extraneous root since log 3
is undefined.
9
x x
x x
x x
SS17
2 2 2
2 2 2
2 2 2
2 2
2
2
2
21
6. log 2 1 log 2 log 1 1
log 2 1 log 2 log 1 1
log 2 1 log 1 log 2 1
log 2 1 1 log 2 1
2 1 1log 1
2
2 1log 1
2
2 12
2
x x x
x x x
x x x
x x x
x x
x
x x
x
x x
x
log xbx y y b
18
2 2 2
2
2
2
6. log 2 1 log 2 log 1 1
2 12
2
2 1 2 4
2 3 5 0
2 5 1 0
2 5 0 1 0
51
2
51 is extraneous.
2
x x x
x x
x
x x x
x x
x x
x x
x x
SS
19
End of Chapter 3.3
20
