You can choose almost any number as the base for logarithms and exponentials. Since we have ten fingers we often think of 10x 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 bx and .
In particular, let's
start with 2x. Now you might be tempted to say
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.
by the rules for logarithms.
Applying the chain rule, we have
Method 2. We want to find where y= 2x. 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 logb x ?
We want to find where y = logb x.
But, y = logb x implies Let's implicitly differentiate this equation.
We've just shown that: