đặt a=log2 3 , b=log 5 6 , tính log 15 6 theo a b

* Hướng dẫn giải

Đáp án A

$$\begin{gathered} {\log }_{15}6=\frac{{\log }_{2}6}{{\log }_{2}15}=\frac{1+{\log }_{2}3}{{\log }_{2}3+{\log }_{2}5}(*) \\ \mathrm{Ta} \mathrm{có} \mathrm{b}={\log }_{5}6 \\ \iff \mathrm{b}={\log }_{5}3+{\log }_{5}2 \\ \iff \mathrm{b}=\frac{{\log }_{2}3}{{\log }_{2}5}+{\log }_{5}2 \\ \iff \mathrm{b}=\frac{\mathrm{a}}{{\log }_{2}5}+\frac{1}{{\log }_{2}5}=\frac{\mathrm{a}+1}{{\log }_{2}5} \\ \Rightarrow {\log }_{2}5=\frac{\mathrm{a}+1}{\mathrm{b}} \mathrm{thay} \mathrm{vào} (*) \\ \Rightarrow {\log }_{15}6=\frac{1+\mathrm{a}}{\mathrm{a}+{\displaystyle \frac{\mathrm{a}+1}{\mathrm{b}}}}=\frac{\mathrm{b}(1+\mathrm{a})}{\mathrm{ab}+\mathrm{a}+1}=\frac{\mathrm{b}(1+\mathrm{a})}{\mathrm{a}(\mathrm{b}+1)+1} \end{gathered}$$