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Properties of Logarithms. The Product Rule. Let b , M , and N be positive real numbers with b  1. log b (MN) = log b M + log b N The logarithm of a product is the sum of the logarithms. For example, we can use the product rule to expand ln (4 x ): ln (4 x ) = ln 4 + ln x . æ .

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Presentation Transcript
the product rule
The Product Rule

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

logb (MN) = logbM + logbN

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

æ

ö

M

log

ç

÷

=

log

M

-

log

N

è

ø

b

N

b

b

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.

the power rule
The Power Rule

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

log bM p = p log bM

The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.

text example

Solution

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

= log4 64

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

= 3

Text Example

Write as a single logarithm:

a. log4 2 + log4 32

problems
Problems

Write the following as single logarithms:

the change of base property
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.

example changing base to common logs

Solution Because

Solution Because

This means that

This means that

Example: Changing Base to Common Logs

Use common logarithms to evaluate log5 140.

Example: Changing Base to Natural Logs

Use natural logarithms to evaluate log5 140.