For any number b and positive integer n, we define exponentiation, i.e. b raised to the power n, as follows:
bn = b⋯b = b multiplied by itself n times
We can extend this definition to non-positive integers n as follows:
For example, 23 = 2 ∙ 2 ∙ 2 = 8, 2-3 = 1/8 and 20 = 1
Exponentiation has the following properties:
Where n > 0, we can also define the number a such a multiplied by itself n times is b. We can extend this definition to
where m and n are any integers.
Without getting into all the details, ba is defined for any a, and can be calculated in Excel by b^a. The properties noted above for integer exponents can be extended to any exponents, namely
logba, called the log of a (base b) = the number c such that bc = a. Thus, the log function is the inverse of exponentiation and has the following properties:
In this website we use logs with base = 10 (called log base 10 and written simply as log a) and logs with base e where e is a special constant equal to 2.718282…. The log of a base e is called the natural log of a and is written as ln a.