Section 5.3 Properties of Logarithms

Advanced Algebra

Properties of Logarithms

logb1 = 0 Anything raised to the 0 power is 1.

logbb = 1 Anything raised to the 1st power is that anything

logbbx = x Think about as exponent: bx = bx

Rewrite as a log: logbx = logbx

xb xb log

Common and Natural Log Properties

log 1 = 0

log 10 = 1

log 10x = x

10log x = x x > 0

ln 1 = 0

ln e = 1

ln ex = x

eln x = x if x > 0

Simplifying

4log4

1

310log

31010log

3

Simplify Using Properties

3 2ln e3

2

lne

32

4log3 22

64

4log3 223

2 4log234

)4log( 2

10 x

42 x

More Simplifying

x310log

x3

elnln

1ln

0

More Simplifying

01.log

210log

2

25log5

5log5

1

27log3

21

27log3

21

33 3log

23

3log3

2

3

More Simplifying

35 625log

31

625log5

31

45 5log

34

5log5

3

4

3 More Properties

Product Rule logbMN = logbM + logbN

Quotient Rule

Power Rule

NMN

Mbbb logloglog

MpM bp

b loglog

Write as a sum or difference of logs

4 32log yu

432log yu

43

loglog 2 yu

yu log43log2

5log

4

2

x

5loglog 24

2 x

5loglog4 22 x

283

12log

2

2

xx

xx

74

34log

xx

xx

74log34log xxxx

7log4log3log4log xxxx

7log4log3log4log xxxx

Expand as a sum and difference of logs

Expand the log into a sum or difference of logs

441

log22

2

xxx

xx

22 21

1log

xx

xx

22 2log)1log()1log(log xxxx

2log2)1log()1log(log 2 xxxx

Write the difference as a single log

ba ln3ln2

1

3lnln 21

ba

3

21

lnb

a

3lnb

a

Write the sum as a single log

zyx 333 log8log2log

21

32

33 log8loglog zyx 4

32

33 logloglog zyx 42

3log zxy

To use the product or quotient rules of logs, remember, the bases must be the same.

Combine into a single log

zyx ln33lnln2

32 ln3lnln zyx 3

2

ln3

ln zy

x

y

zx

3ln

32

Condense to a single logarithm

4ln6ln3

1)9ln(

4

1 22 xxx

4ln6ln)9ln( 31

41 22 xxx

4ln6ln9ln 3 24 2 xxx

46

9ln

3 2

4 2

xx

x

Evaluate log45

b

MMb log

loglog

y5log4

54 y

5log4log y

5log4log y

4log

5logy

16.1y

b

MMb ln

lnlog

Change of Base Formula

Evaluate

5log3

1

31log

5log

46.1