Let $f(x)=\frac{\left(\sqrt{x}+3\right) \log \left(\frac{x+3}{4}\right)-\sqrt{x} \log (x)}{\left(\sqrt{x}-1\right)^3 \sqrt{x}}$, where $x>0$ and the $\log$ is the natural logarithm (with base $e$). My question is how to show that it has a unique root.
I tried many ways, such as taking its derivative. However, its derivative appears to be even more complicated and intractable.
I also know that $x=9$ is a root but have no idea how to show its uniqueness.
I also plotted the function, however, this is not a proof anyway.
Your graph is a good hint. As $x$ gets very large the denominator always is positive, so you just need to prove that the numerator stays negative. $$(\sqrt x + 3)\log\left(\frac {x+3}4 \right)-\sqrt x \log x=(\sqrt x+3)(\log (x+3)-\log 4)-\sqrt x\log x\\=(\sqrt x+3)\log x\left(\log (1+\frac 3x)-\frac {\log 4}{\log x}\right)-\sqrt x\log x\\\approx (\sqrt x+3)\log x\left(\frac 3x-\frac {\log 4}{\log x} \right)-\sqrt x\log x\\ \lt 0$$ The denominator is getting large faster which is why the function approaches zero, but it won't get there because the numerator is always negative for $x$ large.