Why does rotating the function $-e^x$ 270 degrees seem to equal $\ln(x)$? I saw an illustration in my text book of common functions plotted on cartesian coordinate system, when I noticed a relationship between $e^x$ and $ln(x)$.
I noticed that $e^x$ looked awfully similar to $ln(x)$, and in fact if you rotate $-1 \times e^x $ 270 degrees, the functions seem identical, and I can't put my finger on why that is.
I know the two functions are closely related, but I have never been able see the relationship visually before.
Why is this this the case? Are there any ways to prove this? 
 A: Rotating the relation $y=f(x)$ by 270° turns $y$ to $x$ and $x$ to $-y$, so the relation becomes $f(-y)=x$.
A: The two functions $e^x$ and $\log x$ are inverses of each other. For any function of $x$ defined by some equation $F(x,y)=0,$ the inverse function is obtained by performing the transformation $x\mapsto y.$ This is equivalent to reflecting the curve $F(x,y)=0$ about the line $y=x.$
So given the curve $y=e^x,$ and flipping about $y=x$ gives $x=e^y,$ which defines $y$ as the logarithm of $x.$ That is, the latter equation is equivalent to $y=\log x.$
A: The inverse of $f(x)=-e^x$ is $g(x)=-ln(-x)$
These two functions are symmetric about the line $y=x$, so that corresponds to refecting $f$ on the line $y=x$.
I'll use matrix notation to keep track of the progress. In this case it would be applying the transformation:
$$\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}$$
Then $\ln(-x)$ and $\ln (-x)$ are symmetric about the $y$ axis, which corresponds to reflecting $\ln(-x)$ on the $y$ axis, or applying the transformation:
$$\begin{pmatrix} -1 & 0 \\ 0 & 1\end{pmatrix}$$
So in the end to get from $-e^x$ to $\ln x$ is the successive transformation:
$\begin{pmatrix} -1 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}=\begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}$
Or:
$$\boxed{\begin{pmatrix} \cos (-270^\circ) & -\sin (-270^\circ) \\ \sin (-270^\circ) & \cos(-270^\circ)\end{pmatrix}}$$
NB: The negatives imply clockwise rotation.
