1. Properties of Logarithms
Section 3.3

2. inverses “undo” each each other
Since logs and exponentials of the same base are inverse functions of each other they “undo” each other.
Remember that:
This means that:
inverses “undo” each each other
= 7
= 5

3. = = = = Properties of Logarithms 1. 2. 3. CONDENSED EXPANDED
(these properties are based on rules of exponents since logs = exponents)

4. Misconceptions! log (a + b) NOT the same as log a + log b
log (1 / a) NOT the same as 1 ⁄ (log a)

5. Using the log properties, write the expression as a sum and/or difference of logs (expand).
When working with logs, re-write any radicals as rational exponents.
using the second property:
using the first property:
using the third property:

6. using the third property:
Using the log properties, write the expression as a single logarithm (condense).
using the third property:
this direction
using the second property:
this direction

7. Change-of-Base Formula
Example for
TI-84 Plus
The base you change to can be any base, so generally we’ll want to change to a base so we can use our calculator. That would be either base 10 or base e.
“common” log base 10
LOG LN
“natural” log base e

8. Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer to three decimal places.
Since 32 = 9 and 33 = 27, our answer of what exponent to put on 3 to get it to equal 16 will be something between 2 and 3.
put in calculator

9. More Properties of Logarithms
This one says - if you have an equation, you can take the log of both sides and the equality still holds.
This one says - if you have an equation and each side has a log of the same base, you know the expressions you are taking the logs of are equal.