An exponential function is a function of the form

where *b* is a positive constant and x is any real number. The number
*b* is often called the base. The special
case *b*=1 is usually excluded, because that would just give the
constant function *f*(*x*)=1.

Defining an exponential function in Maple is straightforward. For example, to define and plot the exponential function

the following Maple commands could be used. They also show how to
obtain numerical values for several *x* values.

> f := x -> 2^x;

> f(0);

> f(2.5);

> f(Pi);

> evalf(f(Pi));

> plot(f(x),x=-3..3);

The behavior of an exponential function depends very much on whether
the base, *b* is smaller or larger than 1. For example, look at the
plot generated by the following command. Make sure you understand
which curve is which, and how these two curves are related.

> plot({2^x,(1/2)^x},x=-3..3);

A problem that often arises in applications is to fit data to an
exponential function. For example, suppose you knew that *g*(*x*) was an
exponential function and that . You can use this
information to solve for the value of *b*, because it must satisfy the
equation

In Maple, this can be done with the `solve` or `fsolve`
command. The following example shows how to find the value of *b*, and
use it to construct the function *g*(*x*). Notice the use of the label
`b1` in the definition of the function.

> b1 := solve(b^2.5 = 1/Pi,b);

> g := x -> b1^x;

> g(2.5);

> plot(g(x),x=0..6);

You might ask why it wouldn't have been simpler to just use `b`
for the label. The answer is that this would have worked the first
time you executed the command, but would generate an error the second
time you tried it, as shown below.

> b := solve(b^2.5 = 1/Pi,b);

> b := solve(b^2.5 = 1/Pi,b);

Error, (in solve) a constant is invalid as a variable, .6326158238

The problem the second time is that the label `b` has a value
after the first command above, so it can't act as a variable in the
second command.

The conclusion you should draw here is the following. If you need to
solve an equation in Maple and want to label the result so you can
use it later, **don't** use the same name for the variable in the
equation and the label. Before you go on, you should clear out the
value of `b` with the command below. This makes `b` back
into a variable.

> b := 'b';

Fri Jan 31 14:58:36 EST 1997