​if log5 to the base root 3=p and log2 to the base 3 =q,then find value of log300 to the base 3   ​

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ {\log }_{\sqrt{3}}5=\mathrm{p} \\ \Rightarrow {\log }_{3^{\frac{1}{2}}}5=\mathrm{p} \\ \Rightarrow 2{\log }_{3}5=\mathrm{p} \left[{\log }_{{\mathrm{a}}^{\mathrm{n}}}\mathrm{m}=\frac{1}{\mathrm{n}}{\log }_{\mathrm{a}}\mathrm{m}\right] \\ \Rightarrow {\log }_{3}5=\frac{\mathrm{p}}{2} \\ {\log }_{3}2=\mathrm{q} \\ {\log }_3\left(300\right)={\log }_3\left(3\times 5^2\times 2^2\right)={\log }_3\left(3\right)+{\log }_3\left(5^2\right)+{\log }_3\left(2^2\right) \\ =1+2{\log }_{3}5+2{\log }_{3}2 \\ =1+2.\frac{\mathrm{p}}{2}+2\mathrm{q} \\ =1+\mathrm{p}+2\mathrm{q} \left(\mathrm{Answer}\right) \end{gathered}$$

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