We have

For all and in **R**,

If we apply to both sides of this equality we get

For all we have

from which it follows that

If and we have

If we apply to both sides of this identity we get

In particular,

Now we have defined for all
, but we have only defined
when
and
. (We know what
is,
but we have not defined .)
Because of relation (14.30) we often write in place of .
is
called the *exponential function*, and is written

We can summarize the results of this example in the following theorem:

Proof: We have proved all of these properties except for relation (14.32). The proof of (14.32) is the next exercise.