Tradition dictates the inverse function for be written and be called the “Natural Logarithm”.
Let , we find the derivative to to be
We also see implies . Similarly , we find .
Similarly, since we find .
The remaining exponential identity gives us .
From these statements, we may deduce everything about .
Remark. In practice, we typically consider when performing calculations. The Continued Fraction form [wikipedia.org] gives a quick calculation when combined with the logarithmic identities.
2. Limit Definition. Recall we had the exponential map defined as a limit as
If we let
then we observe that
subtracting 1 from both sides and multiplying through by yields
Taking we get
However, we are fudging a little, but the general idea is correct.
3. Tricky Derivative. Consider the function
What is its derivative? We can rewrite it as
How to find this derivative? We use the chain rule!
After some computation we find
We couldn’t have done it without the natural logarithm!
Exercise 1. What’s the derivative of with respect to ?