Logarithmic differentiation is used in mathematics when dealing with infinite products in complex analysis. Note the construction of $f’/f$ from $f$ does not require logarithms of $f$ as a middle step (except intuitively), so it is perfectly fine to use even when $f$ takes non-positive values. ItThe prime number theorem and the link between prime numbers and the Riemann hypothesis involve the logarithmic derivative of the Riemann zeta-function. That is all too advanced for your students. The topic of logarithmic differentiation feels like an archaic topic in a calculus course, however, since there are almost no worthwhile problems to use it on. Feynman liked to use it to differentiate rational functions with numerator and denominator in factored form. I do not think your students would find this interesting at all.
I think time spent in a calculus course on $x^x$ or the “most general” case of $f(x)^{g(x)}$ is not time well spent. These are worthless things for students to spend time on in calculus when there are so many more important things to discuss. The exponential-type functions that matter are $x^c$ and $c^x$ for constant $c$. You can branch out and use the chain rule on $e^{-x^2}$, which matters in probability, but $x^x$ is a time sink. That $x\log x$ shows up in math (especially for integers only!) is not a compelling argument to care about $x^x$ for a real variable $x$: a function $f(x)$ is not important just because its logarithm is important. Otherwise we’d make a big deal about $e^{\sin x}$ and in basic courses that never shows up in practical situations. The function of a real variable $x^x$ is a curiosity and that is all.