Working with other Bases

You can choose almost any number as the base for logarithms and exponentials.
Since we have ten fingers we often think of 10^{x} and the corresponding
logarithm function
.
In the old days, schoolchildren got lots of
practice using tables of logarithms with base ten. If we had two fingers we
might have used base two. Come to think of it, we do have two arms.
Anyway, these days we only use one finger to peck at the keys of a calculator
and the function
is disappearing from the world.

Let's look at the derivatives of *b*^{x} and .
In particular, let's
start with 2^{x}. Now you might be tempted to say

You have to learn to control your baser instincts. The power rule doesn't apply when the base is a constant and the exponent is a variable. Instead, the correct answer is:

A factor of multiplies the derivative. Why , for goodness sake, you ask?

We will show you two different ways to obtain this answer. You can decide which you like better and stick to that one.

Method 1. Well, look.
,
so that
by the rules for logarithms.
Applying the chain rule, we have

Method 2. We want to find
where *y*= 2^{x}. Let's use logarithmic
differentiation on that last equation.

Just what we were expecting.

The same argument in either method gives the general case:

Now, what about differentiating *log*_{b} *x* ?

We want to find
where
*y* = *log*_{b} *x*.

But,
*y* = *log*_{b} *x* implies
Let's implicitly differentiate
this equation.

We get:

We've just shown that:

An extra factor of occurs for exponentials with base