= Differential entropy under an increasing transformation
{title2=$h_b(g(X))=h_b(X)+\mathbb E\log_b g'(X)$}
Let $X$ have a <probability density function> on an open interval, and let $g$ be continuously <differentiable>, strictly increasing there, with $g'(x)>0$. The <change of variables formula> gives
$$
f_{g(X)}(g(x))=\frac{f_X(x)}{g'(x)}.
$$
If the <differential entropy> and the expected absolute <logarithm> of $g'(X)$ are finite, taking the <expectation> of the negative <logarithm> proves the displayed formula, for a fixed base $b>1$. Local stretching reduces <probability density>; its contribution to <differential entropy> is the expected logarithmic stretch. The map is invertible, so this change does not mean that the transformation destroyed information about $X$.
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