3.3 Properties of Logarithms

Logarithmic Properties

If is a positive number not equal to 1, is a real number, andandare positive real numbers then the following properties are true:

Property 1:

Example:

Property 2:

Example:

Property 3:

Example:

These properties mentioned above are also true for natural logarithms,, under the same conditions, so can be substituted with if needed. These properties can help us to expand and condensing logarithmic expressions.

Expanding Logarithmic Expressions

Expand this expression:

Property 3

Property 2

Property 1

Condensing Expressions

Condense this expression:

   Property 3

Property 1

Property 2

Change-of-Base Formula 

                                        

If you are trying to solve for logarithms that have bases other than those on your calculator (andare the most common on calculators) ,you can use the change -of –base formula to solve for them. 

If  and are positive real numbers and andthen you can change the base ofwith any of these given formulas:

                                                             

    Base changed to b

  Base changed to 10

        Base changed to e

Example:

Base change to 10

               

               

- This process would be the same if you wished to change the base to e or any other positive real number, just replace 10 with the value you want.