1. Properties of Logarithms

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. 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:

5. 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

6. 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 "stuff" you are taking the logs of are equal.

7. There is an answer to this and it must be more than 3 but less than 4, but we can't do this one in our head.
(2 to the what is 8?)
Let's put it equal to x and we'll solve for x.
Change to exponential form.
(2 to the what is 16?)
use log property & take log of both sides (we'll use common log)
(2 to the what is 10?)
use 3rd log property
Check by putting in your calculator (we rounded so it won't be exact)
solve for x by dividing by log 2
use calculator to approximate

8. Change-of-Base Formula
If we generalize the process we just did we come up with the:
Example for TI-83
Change-of-Base Formula
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

9. 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

10. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.
Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar