G4.3 If $2^{m}=3^{n}=36$ and $R=\frac{1}{m}+\frac{1}{n}$, find the value of $R$.
$\begin{array}{l}\log 2^{m}=\log 3^{n}=\log 36 \\ m \log 2=n \log 3=\log 36 \\ m=\frac{\log 36}{\log 2} ; n=\frac{\log 36}{\log 3} \\ \frac{1}{m}+\frac{1}{n}=\frac{\log 2}{\log 36}+\frac{\log 3}{\log 36}=\frac{\log 6}{\log 36}=\frac{\log 6}{2 \log 6}=\frac{1}{2}\end{array}$