Transcript Properties of Logarithms The Product Rule Let b, M, and N be positive real numbers with b  1. logb (MN) = logb M.

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.
M 
log b   = log b M - logb N
 N 
The logarithm of a quotient is the
difference of the logarithms.
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 Example
Write as a single logarithm:
a. log4 2 + log4 32
Solution
a. log4 2 + log4 32 = log4 (2 • 32)
= log4 64
=3
Use the product rule.
Although we have a single logarithm,
we can simplify since 43 = 64.
Problems
Write the following as single logarithms:
a.log 25  log 4
b.log  7 x  6  - log x
1
c. log x  4 log  x - 1
2
d .3ln  x  7  - ln x
The Change-of-Base Property
For any logarithmic bases a and b, and
any positive number M,
loga M
logb M =
loga b
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
Use common logarithms to evaluate log5 140.
Solution Because logb M =
log5 140 =
log a M
,
log a b
log140
 3.07.
log5
This means that  3.07.
Example: Changing Base to Natural Logs
Use natural logarithms to evaluate log5 140.
Solution Because logb M =
log5 140 =
log a M
,
log a b
ln140
 3.07.
ln5
This means that  3.07.
Example
Use logarithms to evaluate log37.
Solution:
log 7
log3 7 =
10
log10 3
or
ln 7
log 3 7 =
ln 3
so
log3 7 = 1.77