Page 173 - Calculus Demystified

Transcendental Functions
CHAPTER 6
160
You Try It: Use your calculator to compute log e and ln 10 = log 10 (see
e
10
Example 6.20 below). Confirm that these numbers are reciprocals of each other.
6.3 Exponentials with Arbitrary Bases

6.3.1 ARBITRARY POWERS

We know how to define integer powers of real numbers. For instance
1 1
4
6 = 6 · 6 · 6 · 6 = 1296 and 9 −3 = = .
9 · 9 · 9 729
But what does it mean to calculate
e
4 π or π ?
You can calculate values for these numbers by punching suitable buttons on your
calculator, but that does not explain what the numbers mean or how the calcu-
lator was programmed to calculate them. We will use our understanding of the
exponential and logarithm functions to now define these exponential expressions.
If a> 0 and b is any real number then we define

b
a = exp(b · ln a). (∗)
To come to grips with this rather abstract formulation, we begin to examine some
properties of this new notion of exponentiation:
If a is a positive number and b is any real number then
b
(1) ln(a ) = b · ln a.
In fact
b
ln(a ) = ln(exp(b · ln a)).
But ln and exp are inverse, so that the last expression simplifies to b · ln a.

EXAMPLE 6.14
4
Let a> 0. Compare the new definition of a with the more elementary
4
definition of a in termsof multiplying a by itself four times.
SOLUTION
4
We ordinarily think of a as meaning
a · a · a · a.