Effect of Log change of scale on $\delta$ distribution

Question

I am aware of the multiplicative adjustment when dilating a $\delta$-distribution: $$\delta(ax)=\frac{1}{a}\delta(x), a>0 $$

Suppose we would like to map $\mathbb{R}\to\mathbb{R}_+$ under $x\to\log(x)$ how does $\delta$ change under the change?

As the origin maps to $x=1$ we should have a $\delta$ centered at $x=1$, but is there also a multiplicative adjustment? thanks

Answer 1

For a differentiable function $f$ with zeros $x_1,\ldots,x_n$, i.e. $f(x_1)=\ldots=f(x_n)=0$, the following equation holds:

$$\delta(f(x)) = \sum_{i=1}^n\frac{1}{|f'(x_n)|} \delta(x-x_n)$$

(See also on Wikipedia)

Thus,

$$\delta(\log(x)) = \frac{1}{|\log'(1)|}\delta(x-1)$$