exponential and logarithmic equations section 3.4
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Solve the equation
345 x
x
x
x
5log34log
)34(log
345
5 OR
5log34log
34log5log
34log5log
x
x
x
5ln34ln
34ln5ln
34ln5ln
x
x
x
OR
Take the logarithm of both sides of the equation
Change to logarithmic form
Solve the equation
2041 xe
x
x
x
e x
4120ln
4120ln
4120ln
2041
4
120ln
120ln4
20ln41
20lnln41
20lnln
2041
41
x
x
x
ex
e
ex
x
OR
Change to logarithmic form
Take the logarithm of both sides of the equation
Solve the equation0202 xx ee
045
0202
xx
xx
ee
ee
Change to logarithmic form
x
e
ex
x
)4ln(
4
04
x
e
ex
x
)5ln(
5
05
OR
Take the logarithm of both sides of the equation
4ln)ln(
4
04
x
x
x
e
e
e
5ln
)5ln(ln
)5ln(ln
5
05
x
ex
e
e
e
x
x
x
OR
negative numbers are not in the domain of a logarithm
negative numbers are not in the domain of a logarithm
Solve using factoring
Solve the equation0202 xx ee
42
91OR5
291
291
2811
28011
1*2)20)(1(411 2
xx
x
x
x
x
ee
e
e
e
e
Change to logarithmic form
x
ex
)4ln(
4
x
ex
)5ln(
5OR
Take the logarithm of both sides of the equation
4ln)ln(
4
x
x
e
e
5ln
)5ln(ln
)5ln(ln
5
x
ex
e
ex
x
OR
negative numbers are not in the domain of a logarithm
negative numbers are not in the domain of a logarithm
Solve using the quadratic formula
• What is the initial number of bacteria?
• What is the relative growth rate of the bacterium population
The number of bacteria in a culture is modeled bywhere t is in hours.
tetn 54.02310)(
1*2310)0(
2310)0(
2310)0(0
0*54.0
n
en
en
Initial population is 2310 bacteria.
The relative growth rate is .54 or 54%.
• How many bacteria will there be in three hours?
The number of bacteria in a culture is modeled bywhere t is in hours.
tetn 54.02310)(
63863.11672)3(
2310)3( 3*54.0
n
en
The population in three hours will be 11673 bacteria. Note: 11672 bacteria would also be accepted.
• How many hours will it take for there to be 10000 bacteria?
The number of bacteria in a culture is modeled bywhere t is in hours.
tetn 54.02310)(
It will take 2.713589 hours for there to be 10000 bacteria.
t
t
t
e
e
t
t
71358809.254.231010000
ln
54.231010000
ln
231010000
231010000
54.0
54.0
Solve the equation
0)3ln(17 x
xe
xe
xe
x
3
3
3
)3ln(17
17
17
17
xe
xe
xe
ee
xx
3
3
3
)3ln(17
17
17
17
)3ln(17
Change to exponential form
OR
Exponentiate both sides of the equation
Solve the equation0)8ln()8ln( xx
064ln
088ln
0)8ln()8ln(
2
x
xx
xx
x
x
x
xe
65
65
641
64
2
2
20
Change to exponential form
OR
Exponentiate both sides of the equation
65OR65
065OR065
06565
065
1642
2
064ln 2
xx
xx
xx
x
x
ee x
Solve the equation0)8ln()8ln( xx
00622577483.ln06225775.16ln
0865ln865ln
0865ln0622577483.ln
0865ln865ln
Check possible solutions in original equation
Continued
negative numbers are not in the domain of a logarithm
arguments are both positive
only solution is 65
Solve the equation
5OR10
50OR100
5100
5050
4654
4652
2
2
22
xx
xx
xx
xx
xx
xx
2465log 22 xx
5OR10
05OR010
0510
0505
4465
22
2
2
2465log 22
xx
xx
xx
xx
xx
xx
Change to exponential form
Exponentiate both sides of the equation
OR
Factoring
Check answers in original equation
24log
24650100log
246)10(510log
2
2
22
24log
2462525log
246)5(55log
2
2
22
Both answers are good.
Solve the equation 2465log 2
2 xx
5210
2155
OR102
202155
2155
22255
2200255
)1(2)50)(1(4)5(5
5050
4654
4652
2
2
2
22
xx
x
x
x
x
xx
xx
xx
Change to exponential form
Exponentiate both sides of the equation
5210
2155
OR102
202155
2155
22255
2200255
)1(2)50)(1(4)5(5
0505
4465
22
2
2
2
2465log 22
xx
x
x
x
x
xx
xx
xx
Quadratic Formula
OR
Solve the equation 2465log 2
2 xx
Quadratic Formula
Check answers in original equation
24log
24650100log
246)10(510log
2
2
22
24log
2462525log
246)5(55log
2
2
22
Both answers are good.
Continued
Solve the equation
2OR)0(logOR0
02OR05OR0
025
0525
5
2
xxx
xx
xx
xx
x
x
xx
05252 xx xx
2OR0 xx
0 is not in the domain of a logarithm
only solutions are
Solve the equation44202 32 xx
23log3log4420
3log442023log
3log442023log
3log442023log
2023log443log
2023log)44(
2023log
2
2
22
22
22
22
2
442
x
x
xx
xx
xx
xx
xx
Change to logarithmic form
We will assume that the left side is the exponential function
change of base
22log3log
2log3log
4420
x
Solve the equation44202 32 xx
12log22log2044
2log204412log2
2log20442log2
2log20442log2
442log202log2
442log)202(
442log
3
3
33
33
33
33
3
2023
x
x
xx
xx
xx
xx
xx
Change to logarithmic form
We will assume that the right side is the exponential function
change of base
13log2log
2
3log2log
2044
x
Solve the equation44202 32 xx
3log2log23log4420
2log203log443log2log2
2log203log443log2log2
2log203log443log2log2
3log443log2log202log2
3log)44(2log202
3log2log
2
44202
x
x
xx
xx
xx
xx
xx
Take the logarithm of both sides of the equation
Solve the equation
x
x
6log
6log
11
11
611 x
11ln6ln
16log11x
OR
Change to logarithmic form
Take the logarithm of both sides of the equation
change of base
11ln6ln
1
11ln6ln
6ln11ln
6ln11ln
x
x
x
x
Solve the equation
)15log()17log(log xxx
xxx
xxx
15log17log
)15log()17log(log
015
17log
015log17log
xxx
xxx
Move all logarithms to one side and combine using the Laws of Logarithms
Solve the equation)15log()17log(log xxx
32OR0
032OR0
0)32(
032
1517
1517
115
17
1010
015log17log
2
2
01517
log
xx
xx
xx
xx
xxx
xxxx
xx
xxx
xxx
32OR0
032OR0
0)32(
032
1715
171515
171
1517
10
2
2
0
xx
xx
xx
xx
xxx
xxxx
xxx
xx
Change to logarithmic form
Take the logarithm of both sides of the
equation
Move all logarithms to one side and combine using the Laws of Logarithms -
Continued
OR
Solve the equation)15log()17log(log xxx
Check answers in original equation
)0*15log()170log()0log( )480log(15log)32log(
)32*15log()1732log()32log(
Move all logarithms to one side and combine using the Laws of Logarithms -
Continued
0 is not in the domain of a logarithm
only valid answer is x = 32
Solve the equation)15log()17log(log xxx
xxx
xxx
15log17log
)15log()17log(log
Combine logarithms to have a single logarithm on each side
Exponentiate both sides of the equation
32OR0
032OR0
0)32(
032
1517
15)17(
1010
2
2
15log17log
xx
xx
xx
xx
xxx
xxx
xxx