Exponential and Logarithmic Equations Section 3.4

Exponential and Logarithmic Equations

Section 3.4

Objectives• Solve a logarithmic equation. • Solve an exponential equation.

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



Solve the equation0202 xx ee

045

0202

xx

xx

ee

ee


x

e

ex

x

)4ln(

4

04

x

e

ex

x

)5ln(

5

05

OR


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


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


x

ex

)4ln(

4

x

ex

)5ln(

5OR


4ln)ln(

4

x

x

e

e

5ln

)5ln(ln

)5ln(ln

5

x

ex

e

ex

x

OR



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?


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?


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


OR


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


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



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



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


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


We will assume that the left side is the exponential function

change of base

22log3log

2log3log

4420

x


12log22log2044

2log204412log2

2log20442log2

2log20442log2

442log202log2

442log)202(

442log

3

3

33

33

33

33

3

2023

x

x

xx

xx

xx

xx

xx


We will assume that the right side is the exponential function

change of base

13log2log

2

3log2log

2044

x


3log2log23log4420

2log203log443log2log2




3log)44(2log202

3log2log

2

44202

x

x

xx

xx

xx

xx

xx


Solve the equation

x

x

6log

6log

11

11

611 x

11ln6ln

16log11x

OR



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


Take the logarithm of both sides of the

equation

Move all logarithms to one side and combine using the Laws of Logarithms -

Continued

OR



)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


only valid answer is x = 32


xxx

xxx

15log17log

)15log()17log(log

Combine logarithms to have a single logarithm on each side


32OR0

032OR0

0)32(

032

1517

15)17(

1010

2

2

15log17log

xx

xx

xx

xx

xxx

xxx

xxx


Combine logarithms to have a single logarithm on each side – Continued


)0*15log()170log()0log( )480log(15log)32log(

)32*15log()1732log()32log(


only valid answer is x = 32

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Exponential and Logarithmic Equations Section 3.4