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Why is the natural exponential function defined as y= e^x? Why not something like 2^x or 10^x?

I understand the significance of the constant e , as the limit of $1 compounded continuously , and also as the sum of the series x^n/n!

But what is natural about e^x or conversely about natural logarithm.

Everywhere I see people give me a circular definition.

ijustwannacry
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  • As I am sure your references have mentioned, $e^x$ is the unique solution to $f(0)=1, f'(x)=f(x)$. – lulu Jan 30 '20 at 18:55
  • @lulu not necessarily relevant, but i think it's worth noting that (1+1/n)^n is directly tied in with newton's method with step size 1/n for the differential equation – Saketh Malyala Jan 30 '20 at 18:57
  • @lulu yes I know that , but how does it relate to growth and all? – ijustwannacry Jan 30 '20 at 19:01
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    The differential equation in question is a fundamental growth equation. In words, it describes functions whose rate of growth is proportional to the value of the function. For the particular example, the constant of proportionality is $1$ but that's just a change of scale. – lulu Jan 30 '20 at 19:17

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The exponential function has many beutiful properties. I think that the "most natural" one is: $f'(x)$ = $f(x)$ and $f(0)$ = $1$.

Amit Keinan
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Consider the general function $f(x)= a^x$ where a is any positive number. Then $f(x+ h)- f(x)= a^{x+h}- a^x= (a^x)(a^h)- a^x= a^x(a^h- 1)$. Then the difference quotient is $a^x\frac{a^h- 1}{h}$ and the derivative is given by $\lim_{h\to 0}\frac{a^h- 1}{h}a^x= \left(\lim_{h\to 0}\frac{a^h- 1}{h}\right)a^x$.

That is, the derivative of $f(x)= a^x$, for a any positive number, is a constant, $C_a$, which depends upon a but not x, times the function itself. In particular that constant is 1 if a= e. That is why f'= f and that is why "e" is important.

user247327
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