For any number *b* and positive integer *n*, we define exponentiation, i.e. *b* **raised to the power** *n*, as follows:

*b ^{n}* =

*b⋯b = b*multiplied by itself

*n*times

We can extend this definition to non-positive integers *n* as follows:

For example, 2^{3} = 2 ∙ 2 ∙ 2 = 8, 2^{-3 }= 1/8 and 2^{0} = 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, *b ^{a} *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

log_{b}a, called the** log of ***a* (**base** *b*) = the number *c* such that *b ^{c}*

*= 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*.