Why the $\log$ is so special? When I first learn about the logarithm function $\log$ or $\ln$. My professor said that $\log x$ is a function that when we derive we get the inverse function $1/x$. This $\log$ becomes very popular and "changes the world". I would ask why the $\log$ function is so special? What is the history behind such function?
I mean there are many other functions that we do not know their antiderivatives, e.g., $e^{x^2}$. Why we do not say that for example a function called '$\textrm{something}$' and when we derive it we get $e^{x^2}$. May be I give the bad example here because $e^{x^2}$ has something to do with Gauss but I mean why exactly $1/x$, $\log x$, and $\exp x$?

Not long ago I saw a graph showing the price of bitcoin over the past four years or so (currently in the $\$400\text{--}\$500$ range; it was between $\$2$ and $\$3$ in the fall of 2011).  It looked like this:
\begin{array}{r}
5000 \\[20pt]
500 \\[20pt]
50 \\[20pt]
5 \\[20pt]
0.50 \\[20pt]
0.05 \\[20pt]
& & 2009 &  \qquad\qquad & 2010 & \qquad\qquad &2011 & \qquad\qquad & 2012 & \qquad\qquad & 2013 
\end{array}
. . . with a curve, which you don't see here.  Every time the price gets multiplied by $10$, a certain amound gets added to the height of the curve above the horizontal axis.  In otherwords, the hieght on the vertical axis is a logarithm of the price.
It makes sense to do things this way, since if you buy $\$1$ worth of this commodity at any time, then its dollar-value at a later time depends only on how many times it got multiplied by $10$.  Now suppose it went from $\$0.05$ to $\$0.50$ over some early six-month period, and in some much later six-month period of the same number of weeks it went from $\$50$ to $\$500$.  In each case the logarithm increased by $1$ in the same time.  How fast did the logarithm change?  The rate was $1$ unit per six months.  In both cases.  How fast did the price change?  The rate was $\$0.45$ cents per six months in the first case and $\$450$ per six months in the second case.  Multiplying each by the reciprocal of the size of the price increase, we get $\$0.45\times\dfrac{1}{\$0.45}$ in the first case, i.e. $1$ unit per six months, and in the second case we get $\$450\times\dfrac{1}{\$450}$, again $1$ unit per six months.  I.e. the rate at which the logarithm of the price changes is the rate of change of price times the reciprocal of the price.  That's where $(d/dx)\log x = 1/x$ shows up.
A full answer to the question would be at least 100 times this long; this is just one example.
