3.3 Properties of Logarithms

Properties of Logarithms 

Changing Bases

Changing the base of a logarithm is very useful for solving logarithms on a calculator or . Most calculators only can solve common logs with a base of 10 or natural logs with a base e. 

The equation  

 

is used to change the base of a log.

In the equation c is the new base and a is the old base. 

Ex.1

In this case the base of 3 was changed to base 10.

This can then be solved on a calculator.

*Remember that the old base of 3 goes in the denominator and 243 goes in the numerator

Adding Logarithms 

Logarithms that share a same base and are being added together can be rewritten as  

.

Ex.

By separating the equations and solving them both, it is seen that they both end with the same solution.

                            

   

                  

                                 

Subtracting Logarithms 

Logarithms that share a same base and are being subtracted can also be written as 

.

However, 

This property only works one way.

Ex. 

       

Logarithms with Exponents 

Some logs have an exponent attached to it. Just as with addition and subtraction there is another way to rewrite the equation.

Ex.