definite integration of a function in terms of a composite function over a log-transformed domain

Question

Let $f(x) = g(w)$, where $w=\log(x)$. Can the definite integral $F(b) - F(a) = \int_a^b f(x) \,dx$ be expressed as an integral involving $g(w)$ over the corresponding log-transformed interval (that is, from $\log(a)$ to $\log(b)$)?

Answer 1

Yes. $\frac {dx} {dw }=e^{w}$ so the integral becomes $\int\limits_{\log a }^{\log b} g(w)e^{w}dw$.