What's Special about ' e ' ?

I know e is defined as

Lt_n->¥ (1 + 1/n)ⁿ


But why the heck it is like a magic number ?? What's Special about it ? {Other than being irrational }

Why is that

ò(1/x).dx = log_e x

Why not Logarithm to any other base ??

The point about e is that

http://161.58.186.97/attachment.php?s=&postid=650460

e^x is the only function that differentiates to make itself, which makes it very important in solving differential equations and calculus in general.

we use this idea for the second part of your question to show

http://161.58.186.97/attachment.php?s=&postid=650462


hope this helps