Definition of Base

When one talks about base in mathematics there are a few different definitions they could be refering to:

Numbering System: The most common numbering system is the one we use in everyday life - the decimal system. This is known as base 10, or decimal. Other popular bases are binary (base 2) and hex (base 16). These systems are very common in computer science.
Logarithm Base: What power do I need to raise some number by to get another number? The number you are raising is called the base in logarithms.
Exponent Base: Simply when you take 4 squared, the 4 is known as the base and the 2 (squared) part is the exponent.

Click the above definitions for a more indepth look at each particular instance of bases.