**1.** So we introduced the Exponential function and considered the notion of inverse functions and their derivatives. But what is the inverse function for ?

Tradition dictates the inverse function for be written and be called the **“Natural Logarithm”**.

Let , we find the derivative to to be

(1)

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

(2)

If we let

(3)

then we observe that

(4a)

subtracting 1 from both sides and multiplying through by yields

(4b)

Taking we get

(4c)

However, we are fudging a little, but the general idea is correct.

**3. Tricky Derivative.** Consider the function

(5)

What is its derivative? We can rewrite it as

(5)

How to find this derivative? We use the chain rule!

Thus

(6)

After some computation we find

(7)

We couldn’t have done it without the natural logarithm!

**Exercise 1.** What’s the derivative of with respect to ?

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