# Exponential and logarithmic functions

The next couple of weeks will be about exponential and logarithmic functions.
Logarithms are particularly useful in many applications of mathematics.
## Exponential functions

A **power function**
is a function *f*
of the form
for some constant *n*
called the **exponent** of the function.
You already know about power functions.
An **exponential function**
is a function *f*
of the form

for some constant *b* called the **base** of the function.
The base should be a positive number,
so that *b*^{x} makes sense for every real number *x*.
If you don't remember any other values of an exponential function,
remember these:

*f*(0) = *b*^{0} = 1,
*f*(1) = *b*^{1} = *b*,
*f*(−1) =
*b*^{−1} = 1/*b*.

The domain of an exponential function is the set of all real numbers;
as long as *b* ≠ 1,
the range is the set of all *positive* numbers.
(Because *b* is positive, *b*^{x} is also positive.)
If *b* > 1, then the exponential function is increasing;
if *b* < 1, then the exponential function is decreasing;
and if *b* = 1, then the exponential function is constant.
## Logarithmic functions

As long as *b* ≠ 1,
the exponential function with base *b* is one-to-one,
so it has an inverse.
A **logarithmic function**
is an inverse of an exponential function.
These two statements mean exactly the same thing:
*b*^{x} = *y*;
- log
_{b} *y* = *x*.

If you don't remember any other values of a logarithmic function,
remember these:

- log
_{b} 1 = 0,
- log
_{b} *b* = 1,
- log
_{b} (1/*b*) = −1.

The domain of a logarithmic function is the set of all positive numbers;
the range is the set of all real numbers.
(A logarithm of a negative number is imaginary.)
If *b* > 1, then the exponential function is increasing;
if *b* < 1, then the exponential function is decreasing.
There are abbreviations for logarithms with certain special bases:

- lb
*x* = log_{2} *x*;
- lg
*x* = log_{10} *x*;
- ln
*x* = log_{e} *x*,
where e is a special number, about 2.72;
- log
*x*
is the logarithm of *x* with whatever is your favourite base.

The book's favourite base is 10, which I will also use.

Go back to the course homepage.

This web page was written in 2011 by Toby Bartels,
last edited on 2011 May 11.
Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2011s/logs/`

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