Deriving and defining $e^x$, $\log_b(x)$, $\ln(x)$, and their derivatives?

Question

I am trying to understand the definition and evaluation of $e^x$ and $\log(x)$ and $ln(x)$ and their derivatives, but I can't help but feel that a lot of this stuff is circular. Every time I google this subject or ask about it I feel like people give explanations that depend on one of these other concepts, but then when asking about those other concepts, they define it in terms of the very thing I was originally asking about. I admit I'm finding this frustrating.

I'm trying to start from one piece and move onto the next without assuming knowledge or already pre-supposing the result.

Assume that we have this prior knowledge:

A derivative for function $f(x)$ is defined as:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

For $e$, pretend we've defined this from the compound interest problem but we haven't actually evaluated it yet:

$$e = \lim_{n \to \infty} \left( 1 + \frac{1}{n}\right)^n$$

Okay:

1. How do we evaluate $e$, so we can at the very least define something like $\ln(x) = \log_e(x)$.

2. How do we show that when $f(x) = e^x$, then $f'(x) = e^x$?

3. How do we show that when $f(x) = \log_b(x)$, then $f'(x) = \frac{1}{x \ln(b)}$?

I realize this is a three-part question but if I ask any one of these in isolation people will inevitably answer them in a circular way which I am trying to avoid.

Answer 1

From $$e = \lim_{n \to \infty} \left( 1 + \frac{1}{n}\right)^n$$

you may define

$$e^x = \lim_{n \to \infty} \left( 1 + \frac{x}{n}\right)^n$$

and define $ ln x $ as the inverse function of $e^x$.

For finding derivative of of $e^x$ you need properties of $e^x$ and for derivative of ln x you may use formula for derivative of inverse functions.

The other approach is to start with definition of lnx as an integral $$ lnx = \int _1^x \frac {1}{t} dt$$ and define $e^x$ as the inverse function of $lnx$.

This way makes derivatives almost trivial.

Answer 2

So one of the problems you've got is that just defining the number $e$ doesn't get you very far towards defining $e^x$; after all, defining $e^x$ for irrational $x$ might require some sort of definition on the rationals and an extension by continuity. Probably not going to be as insightful as you might hope.

So I'd suggest going in the complete opposite direction.

Step 1. Define $\log x = \int_1^x \frac 1 t \, dt$. Then clearly $\log x$ is a monotonically increasing, continuous function whose derivative is $1/x$. All the usual properties of $\log$, such as $\log(ab) = \log a + \log b$, follow easily from changes of variables and algebra.

Step 2. Since $\log$ is monotone, it has an inverse, which we call $\exp$. We now define $e$ to be the unique real number such that $\log e = 1$. The inverse function theorem guarantees that $\exp' = \exp$, just as an application of the derivative of the logarithm.

Step 3. Notice that recasting all the usual logarithm properties through the inverse function gives us all the classical properties of exponentials, such as $exp(a + b) = \exp a \cdot \exp b$.

Now it's easy to define exponentials and logs with other bases.

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This is the way that's done in a lot of calculus textbooks. The far easier way to deal with things, however, is to define $\exp$ via its Taylor series and let $\log$ be its inverse. See, for example, the zeroth chapter of Rudin's Real & Complex Analysis for this.