Given a power 
, you can treat a
and n as variables. So far, you have considered a
to be variable and n constant; in this way, you arrived
at the power function y = 
. By the discussion in the preceding sections,
is well
defined for arbitrary n. Therefore you can now let a
be constant and n vary, whence you obtain the function
Since the independent variable is an exponent, this expression is called the exponential function. Let a > 0 and, of course,
be the principal value of the root.
We will now plot the
exponential function 
.
To start with, compute y for 
integer x; you will
have some difficulties, when x is a fraction! But you
can get around this problem and compute a table of sufficiently
many value pairs by taking square roots.
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You continue in this manner and find the 8th and 16th root. By multiplication of 21/4, 21/2, 23/4 by 2, you can obtain the values of 21/4·2 = 21+1/4, 21/2·2 = 21½,, etc.
The graphs of y = 2x and y = 10x are shown. The x-axis is the asymptote of both of them.
All curves 
pass
through the point P(0,1),
because a0 has for all a the value 1.
For a =
1, exhibits something peculiar. Since y
= 1x equals 1 for any value of x or,
in other words, for a = 1, arbitrarily many values of x
are associated with y = 1 . We conclude
therefore that (a > 0 and 
1) is always a single-valued
function of x and
conversely:
Every
positive number y can
be represented as a power with an arbitrary base a
(a>0 and 1).