ACTIVITY 38

Laws of Logarithms (Section 5.3, pp. 409-413)

Laws of Logarithms:

Let a be a positive number, with a ≠ 1. Let A, B, and C be any real numbers with A > 0 and B > 0.

BAAB aaa logloglog 1.

BAB

Aaaa logloglog 2.

ACA aC

a loglog 3.

Example 1:

Evaluate each expression:

95 5log

BAAB aaa logloglog 1.

BAB

Aaaa logloglog 2.

ACA aC

a loglog 3.

0)1(log .1 a

1)(log .2 aa

xa xa )(log .3 xa xa log 4.

9

2log7log 33 2*7log3 14log3

2log216log 33 233 2log16log

4log16log 33

4

16log3

4log3

200

lnln ee 200

ln ee 200e

xex )ln(

200ln e 200

50log18log100log 333

50log18log100log 333

50log18

100log 33

50log

9

50log 33

BAAB aaa logloglog 1.

BAB

Aaaa logloglog 2.

ACA aC

a loglog 3.

0)1(log .1 a

1)(log .2 aa

xa xa )(log .3 xa xa log 4.

509

50

log3

50

1

9

50log3

9

1log3

23 3

1log

23 3log 2

Example 2:

Use the Laws of Logarithms to expand each expression:

x2log2 x22 log2log x2log1

xx 54log 25 xx 54loglog 5

25

xx 54loglog2 55

BAAB aaa logloglog 1.

BAB

Aaaa logloglog 2.

ACA aC

a loglog 3.

0)1(log .1 a

1)(log .2 aa

xa xa )(log .3 xa xa log 4.

z

yx5log

z

yx 55 loglog

2

1

55 loglogz

yx

z

yx 55 log

2

1log xyx 555 loglog

2

1log

BAAB aaa logloglog 1.

BAB

Aaaa logloglog 2.

ACA aC

a loglog 3.

0)1(log .1 a

1)(log .2 aa

xa xa )(log .3 xa xa log 4.

xyx 555 log2

1log

2

1log

Example 3:

BAAB aaa logloglog 1.

BAB

Aaaa logloglog 2.

ACA aC

a loglog 3.

0)1(log .1 a

1)(log .2 aa

xa xa )(log .3 xa xa log 4.

srdcb aaa logloglog

Use the Laws of Logarithms to combine the expression into a single logarithm.

rac

aa sdb logloglog

rac

a sbd loglog

r

c

a s

bdlog

Example 4:

Use the Laws of Logarithms to combine the expression into a single logarithm.

xxxx ln4ln352ln2

11ln5ln

xxxx ln4ln52ln1ln5ln 32

1

xxxx ln4ln52ln1ln5ln 3

xxxx ln4ln52ln15ln 3

xxxx ln4ln5215ln 3

xxxx ln4ln5215ln 3

xx

xxln

4

5215ln 3

x

x

xx34

5215

ln

xx

xx 1

4

5215ln 3

xx

xx34

5215ln

Example 5 (Forgetting):

Ebbinghaus’s Law of Forgetting states that if a task is learned at a performance level P0, then after a time interval t the performance level P satisfies

1logloglog 0 tcPPwhere c is a constant that depends on the type of task and t ismeasured in months.

(a) Solve the equation for P.

1logloglog 0 tcPP

BAAB aaa logloglog 1.

BAB

Aaaa logloglog 2.

ACA aC

a loglog 3.

0)1(log .1 a

1)(log .2 aa

xa xa )(log .3 xa xa log 4.

ctPP 1logloglog 0

ct

PP

1loglog 0

ct

P

P 1log

log0

1010

ct

PP

10

(b) Use Ebbinghaus’s Law of Forgetting to estimate a student’s score on a biology test two years after he got a score of 80 on a test covering the same material. Assume c = 0.3.

ct

PP

10

3.0124

80

626527804.2

80 46.30

800 P24t

3.0c

Example 6 (Biodiversity):

Some biologists model the number of species S in a fixed area A (such as an island) by the Species-Area relationship

where c and k are positive constants that depend on the type of species and habitat.

,logloglog AkcS

(a) Solve the equation for S.

kAcS logloglog

AkcS logloglog

kcAS loglog

kcAS loglog 1010 kcAS

(b) Use part (a) to show that if k = 3 then doubling the areaincreases the number of species eightfold.

kcAS

3k

3cAS Let the area be A0

Then double the area is 2A0

301 2AcS

Number of species when the Area is A0 is

300 AcS Number of species when the Area is 2A0

308 Ac 308 Ac 08S

Change of Base:

For some purposes, we find it useful to change from logarithms in one base to logarithms in another base. One can prove that:

bx

xa

ab log

loglog

Example 7:

Use the Change of Base Formula and common or natural logarithms to evaluate each logarithm, correct up to five decimal places:

2log5 )5ln(

)2ln(430676558.

bx

xa

ab log

loglog

125log4)4ln(

)125ln(482892142.3

5log3 )3ln(

)5ln(929947041.2