Properties of Logarithms

The Product Rule

• Let b, M, and N be positive real numbers with b 1.

• logb (MN) = logb M + logb N• The logarithm of a product is the sum of the

logarithms. • For example, we can use the product rule to

expand ln (4x): ln (4x) = ln 4 + ln x.

The Quotient Rule

• Let b, M and N be positive real numbers with b 1.

• The logarithm of a quotient is the difference of the logarithms.

logb

M

N

⎛ ⎝ ⎜

⎞ ⎠ ⎟ =logbM−lobbN

The Power Rule

• Let b, M, and N be positive real numbers with b = 1, and let p be any real number.

• log b M p = p log b M• The logarithm of a number with an

exponent is the product of the exponent and the logarithm of that number.

Text ExampleWrite as a single logarithm:

a. log4 2 + log4 32

Solution

a. log4 2 + log4 32 = log4 (2 • 32) Use the product rule.

= log4 64

= 3

Although we have a single logarithm, we can simplify since 43 = 64.

Properties for Expanding Logarithmic Expressions

• For M > 0 and N > 0:

1. logb (MN) =logbM + logbN

2. logbMN ⎛ ⎝ ⎜

⎞ ⎠ ⎟ =logbM−logbN

3. logbMp =plogbM

Example

3log5log3

5log 2

22

2

2 −= xx

• Use logarithmic properties to expand the expression as much as possible.

Example cont.

3loglog5log

3log5log3

5log

22

22

22

2

2

2

−+=

−=

x

xx

Example cont.

3loglog25log

3loglog5log

3log5log3

5log

222

22

22

22

2

2

2

−+=−+=

−=

xx

xx

Properties for Condensing Logarithmic Expressions

• For M > 0 and N > 0:

1. logb M + logbN =logb(MN)

2. logbM−logbN =logbMN ⎛ ⎝ ⎜

⎞ ⎠ ⎟

3. plogbM =logbMp

The Change-of-Base Property

• For any logarithmic bases a and b, and any positive number M,

• The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base.

b

MM

a

ab log

loglog =

Use logarithms to evaluate log37.Solution:

3log

7log7log

10

103 =

77.17log3 =

3ln

7ln7log3 = or

so

Example

Properties of Logarithms