1. Properties of Logarithms

2. logb (MN) = logb M + logb N
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.

3. The Quotient Rule log M N æ è ç ö ø ÷ = -
Let b, M and N be positive real numbers with b  1.
The logarithm of a quotient is the difference of the logarithms.
log b M N æ è ç ö ø ÷ = -

4. 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.

5. Text Example Write 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.

6. Problems
Write the following as single logarithms:

7. 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.

8. Example: Changing Base to Common Logs
Use common logarithms to evaluate log5 140.
Solution Because
This means that
Example: Changing Base to Natural Logs
Use natural logarithms to evaluate log5 140.
Solution Because
This means that

9. Example
Use logarithms to evaluate log37.
Solution: or so