Solving with Unlike Bases. Warm Ups on the next 3 slides….

Solving with Unlike Bases

Warm Ups on the next 3 slides….

Solve.5

3 1 132

4

x

x

Graph. 42 3xy

Condense. 2 2 23log 4log log 5x y

17

15

1517

102515

22

22102515

52135

x

x

xx

xx

xx

5log

5logloglog43

2

24

23

2

yx

yx

UPWARM

Warm Up

0loglog: 34 xSolve

3

3

1log

log4

1

3

30

x

x

x

x

Warm Up

xSolve 8log:2

1

3

3

22

82

1

3

x

x

x

x

The Common Logarithm

The base-10 logarithm is called the common logarithm.

The common logarithm is usually written as logx.

If we can manage to make everything into log base 10, we can use our calculators to solve.

xx loglog 10

For Example….Solve .625 x

56.2

5log

62log

62log5log

62log5log

625

x

x

x

x

x

Solve: 5 28x

07.2

5log

28log

28log5log

28log5log

285

x

x

x

x

x

Solve: 6.4 59x

20.2

4.6log

59log

59log4.6log

59log4.6log

x

x

x

x

Solve: 8 73x

06.2

8log

73log

73log8log

73log8log

x

x

x

x

Solve:34 45x

92.0

74593.23

4log

45log3

45log4log3

45log4log 3

x

x

x

x

x

Solve: 5 29x

09.2

5log

29log

29log5log

29log5log

x

x

x

x

Solve:73 76x

06.3

73log

76log

3log

76log7

76log3log)7(

76log3log 7

x

x

x

x

x

Solve: 7 2 19x

58.3

2log

12log

12log2log

12log2log

122

x

x

x

x

x

Solve: 21

123

x

13.1

26186.22

3

1log

12log2

12log3

1log2

12log3

1log

2

x

x

x

x

x

Change of Base

There is a way to write equivalent logarithmic expressions with different bases.

There is a formula to find the change of base.

Let’s derive the change of base formula…..

Let’s derive the change of Base Formula!

logb x

b

xy

xby

xb

xb

yx

y

y

b

log

log

loglog

loglog

log

b

xxb log

loglog

Change of Base Formula:

loglog

logb

xx

b

Evaluate to the nearest hundredth.

6log 78

43.26log

78log78log 6


2log 23

52.42log

23log23log 2


1

4

log 35

56.2

4

1log

35log


42 log 17

04.44log

17log2

Natural Log

Warm Up: Find the final amount of money if you invested

$6700 compounded monthly at 5.4% for 20 years.

$4500 compounded continuously at 7% for 20 years.

61.681,19

12

054.016700

1

2012

A

A

n

rPA

nt

40.248,18

4500 2007.0

A

eA

PeA rt

Warm Up: $1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of months required for the value of the investment to exceed $3000.

MONTHSxt

YEARSt

t

t

n

rPA

t

t

t

nt

894.8812

369762662.7

0125.1log

3log12

0125.1log123log

0125.1log3log

0125.13

12

15.0110003000

1

12

12

12

Warm Up:

25.0

.2,1

xgf

equationtheSolvexgandxxfLet x

21

2

1 :1

xxf

xy

yx

xy

xffindst

1

22

22

4

12

25.02

:

22

2

2

x

x

ncompositiothefindNext

x

x

x

Natural Base

e = 2.718281828…

Models a variety of situations of growth & decay in nature

Represented as: ln (means loge)

elogln

Evaluate to the nearest thousandth.

4e

598.54


2.3e

100.0


2

5e

492.1

Evaluate to the nearest thousandth. (Think of 2 Ways…….

ln6

792.16ln

:

DirectlyCalculatorWith

792.1

log

6log

6log

6ln

:1

e

stBaseofChangeUse

e


ln(2.5)

916.0


ln( 2)

.

:

numbersrealPOSITIVEall

isDomainREMEMBER

SolutionNo

!valuedomainNegative

Find

ln( )e

1

log

x

ee

xex

e

Simplify each expression.

ln7e

7

7lnln

7lnlog

7ln

x

x

x

xe

e


ln12e

12

12lnln

12lnlog

12ln

x

x

x

xe

e


2ln3e

9

9lnln

3lnln

3ln2log2

3ln2

x

x

x

x

xe

e

Simplify each expression. (2 Ways)

9lne

9

log

ln

9

9

9

x

ee

xe

xe

x

e or x

x

xe

xe

9

19

ln9

ln 9

Simplify each expression. (2 Ways)46lne

2446

46

6

log6

ln6

4

4

4

x

ee

xe

xe

x

e or

x

x

xe

xe

24

124

ln24

ln64

Solve – Round to the nearest hundredth. (2 Ways)

77.1 x

67.3

7.1log

7log

7log7.1log

7log7.1log

x

x

x

x

or

67.37.1ln

7ln

7ln7.1ln

7ln7.1ln

x

x

x

x

Solve – Round to the nearest hundredth. (Use ln)

8xe

08.2

8ln

8ln1

8lnln

8lnln

x

x

x

ex

e x

Solve – Round to the nearest hundredth.

2 18xe

20.2

9ln

9lnln

9

2

18

2

2

x

x

ex

e

e

x

x


312 48xe

46.03

4ln

4ln3

4lnln3

43

x

x

x

ex

e x


74.5 90xe

00.4

720ln

20ln7

20lnln7

207

x

x

x

ex

e x

Solving with Unlike Bases. Warm Ups on the next 3 slides….

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