Answer
The solution is
$$\lim_{x\to\infty}\frac{\log_2 x}{\log_3 x}=\frac{\ln 3}{\ln 2}.$$
Work Step by Step
We will use the logarithmic rule for base change which says
$$\log_{b}a=\frac{\ln a}{\ln b}.$$
This gives
$$\log_{2}x=\frac{\ln x}{\ln 2},\quad \log_{3}x=\frac{\ln x}{\ln 3}.$$
Putting this into the limit we get
$$\lim_{x\to\infty}\frac{\log_{2}x}{\log_{3}x}=\lim_{x\to\infty}\frac{\frac{\ln x}{\ln 2}}{\frac{\ln x}{\ln 3}}=\lim_{x\to\infty}\frac{\ln 3}{\ln 2}=\frac{\ln 3}{\ln 2}.$$
