Properties of Logarithms. The Product Rule Let b, M, and N be positive real numbers with b 1. log b...
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Transcript of Properties of Logarithms. The Product Rule Let b, M, and N be positive real numbers with b 1. log b...
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