# Exponential and logarithmic functions (§§6.3&6.4)

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

Recall that a **power function**
is a function *f*
of the form
for some constant *n*
called the **exponent** of the function.
In contrast, 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*.
Recall that a **linear function**
is a function *f*
of the form

for some constants
*m* (the *rate of change*)
and *b* (the *initial value*).
Analogously, a **generalized exponential function**
is a function *f*
of the form
for some constants
*b* (the *base*) and *C* (the *initial value*).
If you don't remember any other values of a generalized exponential function,
remember these:

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

The domain of a generalized exponential function
is the set of all real numbers;
as long as *b* ≠ 1,
the range of the exponential function with that base
is the set of all *positive* numbers.
(Because *b* is positive, *b*^{x} is also positive.)
Then the range of a generalized exponential function
is the set of all real numbers with the same sign as *C*.
If *C* > 0 and *b* > 1,
then the generalized exponential function is increasing;
if either of these is reversed, then the function is decreasing;
if both are reversed, then it's increasing again.
(If *b* = 1 or *C* = 0,
then the function is constant.)
Besides numbers such as 10, 2, and 1/2, which you are familiar with,
another common choice of base is a special number, about 2.72, known as e.
The importance of this number e will become clear when we look at applications.
Many calculators have a button
that calculates e^{x} from *x*;
in particular, you can calculate e itself, as e^{1}, using this button.

## 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*,
*b* > 0, and *b* ≠ 1;
- log
_{b} *y* = *x*.

The left-hand side of the latter equation
is the **logarithm**, base *b*, of *y*;
logarithms are particularly useful in many applications of mathematics.
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 logarithmic function is increasing;
if *b* < 1, then the logarithmic 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 the same special number from before, 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.

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This web page was written between 2011 and 2017 by Toby Bartels,
last edited on 2017 May 3.
Toby reserves no legal rights to it.
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