Q:

Change of variables notation clarification

In D. Mackay's book on Information Theory he states that we can make a change of variables for a probability density $P(x)$ by for example $l=\ln(x)$. Then, the density of $x$ is transformed to $$P(l)=P(x(l))\left\vert \frac{\delta x}{\delta l}\right\vert = P(x(l))x(l)$$
I'm confused about the notation of $x$ as a function of $l$: $x(l)$. What exactly is $x(l)$? If I got this right, then we have $l=\ln(x) \rightarrow x = \exp(l) $, so $$P(x(l))\left\vert \frac{\delta x}{\delta l}\right\vert = P(x(l))\left\vert \frac{\delta \exp(l)}{\delta l}\right\vert = P(x(l))\exp(l)=P(x(l))x$$
So are we essentially saying $x(l)=\exp(l)$? Is it that trivial and just a notational trick?

A:

When doing change of variables one typically uses a smooth invertible transformation. So if $l = f(x)$ we can also think of $x = f^{-1}(l)$.
For notational convenience, many authors simply write $x(l)$ instead of $f^{-1}(l)$. That is, yes it is trivial and just a notational trick.

