How to simplify complex logs

Question

I have f(x) = (x+$\sqrt{x}$)$\log_2x$ and g(x) = x$log_2(x+\sqrt{x})$. How would I go about simplifying them and obtaining the limit at infinity of f(x)/g(x). So far, the best I have gotten for f(x) is x$log_2x$(1+$\frac{1}{\sqrt{x}}$), and nothing for g(x).

Any help is appreciated!

Answer 1

As an Idea you can take substitution like $u=\sqrt x$ which makes some simplify $$\lim_{x \to \infty}\frac {f(x)}{g(x)}=\\\lim_{u \to \infty}\frac {(u^2+u)\log_2{u^2}}{u^2\log_2{(u^2+u)}}=\\\lim_{u \to \infty}\frac {2(u+1)\log_2{u}}{u\log_2{(u^2+u)}}= \\\lim_{u \to \infty}\frac {2(u+1)}{u}\lim_{u \to \infty}\frac {\log_2{u}}{\log_2{(u^2+u)}}=\\\lim_{u \to \infty}\frac {2(u+1)}{u}\lim_{u \to \infty}\frac {\log_2{u}}{\log_2{u^2(1+\frac 1u)}}$$ does it helps?

Answer 2

$$\frac{f(x)}{g(x)} =\frac{(x+\sqrt{x})\log_2(x)}{x\log_2({x+\sqrt{x})}}=\frac{\ln\left({x^{x+\sqrt x}}\right)}{\ln\left({(x+\sqrt x)^{x}}\right)}$$