Question Number 197250 by sciencestudentW last updated on 11/Sep/23
$$\mathrm{log}\:\mathrm{3}={a} \\ $$$$\mathrm{log}\:\mathrm{4}={b} \\ $$$$\mathrm{lo}\underset{\mathrm{5}} {\mathrm{g}36}=?\: \\ $$
Answered by AST last updated on 11/Sep/23
$$\frac{{log}\mathrm{3}}{{log}\mathrm{2}+{log}\mathrm{5}}={a} \\ $$$$\frac{\mathrm{2}{log}\mathrm{2}}{{log}\mathrm{2}+{log}\mathrm{5}}={b}\Rightarrow{log}\mathrm{2}\left(\mathrm{2}−{b}\right)={blog}\mathrm{5} \\ $$$${log}_{\mathrm{5}} \mathrm{36}=\frac{\mathrm{2}{log}\mathrm{2}+\mathrm{2}{log}\mathrm{3}}{{log}\mathrm{5}}=\frac{\mathrm{2}{log}\mathrm{2}+\mathrm{2}{alog}\mathrm{2}+\mathrm{2}{alog}\mathrm{5}}{{log}\mathrm{5}} \\ $$$$=\mathrm{2}\left(\frac{{b}}{\mathrm{2}−{b}}\right)+\frac{\mathrm{2}{ab}}{\mathrm{2}−{b}}+\mathrm{2}{a}=\frac{\mathrm{2}{b}}{\mathrm{2}−{b}}+\mathrm{2}{a}\left(\frac{\mathrm{2}}{\mathrm{2}−{b}}\right)=\frac{\mathrm{2}{b}+\mathrm{4}{a}}{\mathrm{2}−{b}} \\ $$