When can we use Taylor series expansion and write $\int_0^{\infty} \log(f(x+\alpha x)) dx = \int_0^{\infty}\log(f(x)+\sum_{n=1}^{\infty}\frac{f^{n}(x) (\alpha x)^n}{n!}) dx$?
I think, first the Taylor expansion should exists, therefore $f$ should be infinitely differentiable. So, if the Taylor series expansion exists, we define $g_k(x)= \log(\sum_{n=1}^{k}\frac{f^{n}(x) (\alpha x)^n}{n!})$ and $g_k(x) \to g(x)= \log(f(x+\alpha x))$ as $k \to \infty$. It is equivalent to see when the following is true
$\lim_{k \to \infty} \int_0^\infty{g_k(x) dx}=\int_0^\infty {g(x) dx} $
If we could find $h(x) > |g_k(x)|$ for all $k$, then we can use the dominated convergence theorem. For the Taylor series and every $f(x)$, does such $h(x)$ exist? Is there a better way to approach this question? What if we couldn't find such $h(x)$?