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