this is a followup question to : Effect of Log change of scale on $\delta$ distribution
the examples that iv'e seen of composition of a delta function with another function somehow use the zeros of the additional function. given that, it makes sense that composing $$\delta( log(x)) = \frac{1}{|log'(1)|}\delta(x-1)$$ where we map $\mathbb{R}\to\mathbb{R}_{>0}$
What is the proper expression if we want to go the other way? suppose we have a shifted delta, $\delta(x-1)$, valid on $\mathbb{R}_{>0}$ and we would like to send $x\to e^x$, what is the proper multiplier on the new delta (and what is the justification)? it makes sense that $\delta(x-1)\to \delta(x-\log(1))=\delta(x)$, but what is the multiplicative adjustment?
thank you