Inverse, Exponential, and Logarithmic Functions

The Number e and the Natural Log

The Number e

e is a number that can be defined in many ways. First,

Also, e is the number such that

The numerical value of e is approximately 2.71828.... The function f (x) = e ^x is an exponential function, which is a function of the form f (x) = a ^x , where a is a positive constant. The graph of f (x) = e ^x is shown below.

The Natural Log

The inverse of the function e ^x is the natural log function ln(x) . These two graphs are pictured below:

Recall that if log _a b = x , then x is the power that a must be raised to in order to equal b .

With the natural log, the base is e . So, for example, ln(e) = 1 . Because e ^x and ln(x) are inverses, the following relations hold:

e ^ln(x) = x and ln(e ^x) = x

Basic Properties of Logarithms

As a review, recall the following basic rules of logarithms:

1. log(MN) = log(M) + log(N)
2. log( = log(M) - log(N)
3. log(M ⁿ) = n(log(M))
4. log _a a = 1
5. log _a1 = 0