Integrating in log-space with a change of variable I have a probability density function $P(f|\mu,\sigma) = \mathcal{N}(f|\mu,\sigma)$.
I need to change the variable $f$ to $L = \log_{10}[f]$ so I can integrate it jointly with another PDF whose domain is in log-space: $=\int P(L|\mu,\sigma)P(L|\theta)dL$.
Think of it as a likelihood: $P(observed|model) = \int P(observed|true) P(true|model)dtrue$.
The domain of the model is $-\infty < L < +\infty \equiv 0 < f < +\infty$ and the domain of the data is $-\infty < f < +\infty$.
So the model is bounded in $f$-space but the data is not (i.e. the data can go negative).
I can transform the data to $L$-space with its jacobian: $P(L|\mu,\sigma) = P(f|\mu,\sigma)f\ln[10]$.
Both functions are well-defined in both spaces (the model just has 0-probability at negative f, so if the data are close to 0, then all that happens is that probability is much less).
But I'm wondering now about the proper use of the Jacobian. It's designed to make sure the PDF integrates to 1. My $P(observed|true)$ will no longer integrate to 1. This is, of course, desired behaviour for the integrand, but is this Jacobian transformation valid? Is this approach correct? I need to integrate in log-space because it is much easier to do so numerically .
I'm struggling to use the right terminology so please excuse lack of knowledge!

If $p(\cdot)$ is a probability density function supported on $\mathbb R^+$ then the probability density function of the (Neper) log-transform is$$\tilde p(y) = p(\exp(y))\exp(y)$$since $$\dfrac{\text d}{\text dy} ~\exp(y)=\exp(y)$$while the probability density function of the decimal -transform is$$\tilde p(y) = p(10^y)\log(10)10^y$$
