Properties of Logarithms

There are three properties of logarithms:

Also, you do not need to only use  ; you can use  , ,   , and more

Here is one way to prove 

First, you can set up the equation:

Then, subtract logb from both sides:

Next, using the definition of a logarithmic function you can change it into this form:

Subtraction in the exponents means you can change it to this:

The denominator can then simplify:

You can multiply both sides by b:

You can then take the log of both sides:

Since , 

You can use these properties to expand or simplify/compress logarithmic expressions:

Expanding:

Simplifying:

There is also the change of base formula:

To prove this formula:

First, you can set up the equation: 

and change it to:

Next, take the log of both sides; it does not have to be base 10:

Using the properties of logarithms:

 

Divide:

 

For example, if you had , you could use the change of base formula to change it to or even . 

One common mistake that people make its they think log(a+b) = loga + logb. This is NOT true!