3.3 Properties of Logarithms

Hello everybody! It's Eleni.
Recently, we've been learning about logarithms.  There are three properties of logarithms.


1. Logarithm Coefficients

Example:

2. Addition

ln (bc) = ln b + ln c
 log_a (bc) = log_a b + log_a c

Example:

Proof of property   log_a (bc) = log_a b + log_a c

1. Make x and y variables

2. Use the key to everything

3. Multiply b and c together

4. Law of exponents

5. Take the log base a of both sides

6. Use the coefficient property of logarithms

7. 

8. Substitute x and y back into equation



3. Subtraction

Example: 

These properties (for math students) are helpful for combining and simplifying logarithmic expressions.  Remember these are the only three. Don't accidentally makeup new ones.  As Wilhelm mentioned in class, if you get confused which property is true, try plugging in some numbers that are easy to work with.

Change of Base Formula

As you have probably figured out, your calculator can not calculate logarithms with any base other than 10 (common log) or e (natural log).

This is what the change of base formula is for.

 Example:

 This you can punch into your calculator and get...

=1.40367...

**NOTE: you can also change the base to the natural log, or whatever is most convenient for the problem.


----Eleni