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My math professor told me that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$ by the definition; so far so good.

But how/why does $\ln(x)$ ($\int_1^x\frac{1}{t} dt$: by defintion) coincide with the inverse of $e^{x}$? Thanks!

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    @aiao: That is not an answer to my question. Analogously, $\log$ is a famous function, but as you can see from your question, it is important to consider precisely how it is defined if you are asking how to prove that it has certain properties. You have not said what $e$ means precisely, and perhaps more importantly, what "to the power $x$" means. E.g., what is $e^e$? (And I don't mean, "What are a few figures from its calculator decimal approximation?".)2012-12-13

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If $f(x) = \int_1^x \dfrac{dt}t$ and $g'(x) = g(x)$ with $g(0) = 1$, then we can show that $f(x) = g^{-1}(x)$.

Setting $t = g(y)$, we get that $dt = g'(y) dy$. When $y=0$, we get that $t=1$ and when $y = g^{-1}(x)$, we get that $t=x$.

Hence, $f(x) = \int_{0}^{g^{-1}(x)} \dfrac{g'(y) dy}{g(y)} = \int_{0}^{g^{-1}(x)} dy = g^{-1}(x)$

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    @aiao $g(0) = 1$. Note that $g'(x) = g(x)$ has infinite functions as solutions unless you specify the value of $g(0)$. $e^x$ is the function such that $g'(x) = g(x)$ and $g(0) = 1$.2012-12-13
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I disagree/disapprove of any math teacher that says this is definition. Its not. No one can assign more than one definition to a single concept without somewhere proving the equivalence.

You want to talk definition? One of the Bernoulli family, while working on problems of bank account growth and compounding interest problems, was able to show that a bank account that grows with 100% APR, compounded continuously, converged to a value a little over 2.7 times that of the principle. HE showed that $\lim_{n\to \infty} (1+\frac{1}{n})^n = e$.

THIS is definition. The most fundamental and principle of equalities. This is the historical origin, the chronological first, and the basis of all other properties which followed, and which were proved equivalent.

So now youve established the existence of e. Taking $e^x$ is a trivial algebraic concept. Taking example from the interest growth problems, $e^x$ is the balance of a bank account with a principle of 1 after $x$ years.

Enter logarithms: If $e^m = n $ then (by definition of the logarithm) we have $m = \log_e(n)$.

Then, we can say $\log_e(n) = \ln(n)$. This is nomenclature. Its just a standard and simpler way of writing a frequently appearing logarithm. This equivalence is not proven, its definition. But its not a mathematical coincidence, its simply an arbitrated truth. It was assigned and invented for simplicity.

Proving $\frac{d}{dx} \ln(x) = \frac{1}{x}$ is done via the limit definition of the derivative. Here, let us prove the derivative of any arbitrary logarithm with respect ot its argument x. $\frac{d}{dx} \log_a(x) = \lim_{h\to 0}\frac{\log_a(x+h)-\log_a(x)}{h}$

$\frac{d}{dx} \log_a(x) = \lim_{h\to 0}\frac{\log_a(\frac{x+h}{x})}{h}$

$\frac{d}{dx} \log_a(x) = \lim_{h\to 0}\frac{1}{h}\log_a(1+\frac{h}{x})$

Letting $h=x/n$ then as $h\to 0$ we have $n\to\infty$. The substitution creates:

$\frac{d}{dx} \log_a(x) = \lim_{n\to \infty}\frac{n}{x}\log_a(1+\frac{1}{n})$

Factoring out constants and continuous functions independent of $n$: $\frac{d}{dx} \log_a(x) = \frac{1}{x}\log_a\lim_{n\to \infty}(1+\frac{1}{n})^n$

By the ACTUAL definition of e, we have $\frac{d}{dx} \log_a(x) = \frac{1}{x}\log_a(e)$

This is a general truth for any log. If you let $a=e$ then $\frac{d}{dx} \ln(x) =\frac{1}{x}\log_e(e) = \frac{1}{x}$. It is because of this that $e$ holds special importance to calculus.

And by the general theorem of calculus, we also have the integral, $\ln(x) = \int_1^x \frac{1}{t} dt$. This is just the reverse rule.

Proving that $\frac{d}{dx}e^x =e^x$ is as easy as letting $y=e^x$ and evaluating $\frac{d}{dx}\ln(y) = \frac{d}{dx}x$ using the chain rule. It falls out kind of trivially.

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    When and if you understand that fact, the question is begged which definition do you have/use? I see no practical reason to switch from the historical and traditional one, wherein students learn the history and conceptual development motivating the need, as they learn the concepts. When you switch definitions, they often become overly complicated and require advanced mathematics to prove theorems what ought to be much simpler for a lower-level student to understand from a more traditional approach. History has developed for us in increasing complexity. Changing that forces over-complication.2018-03-28