Đặt log2 3 = a, log3 15 = b. Biểu thị log 30 18 theo a và b.

Trả lời

Ta có:

$a={\log }_{2}3=\frac{1}{{\log }_{3}2}$$\Rightarrow {\log }_{3}2=\frac{1}{a}$

$b={\log }_{3}15={\log }_3(3.5)$

$={\log }_{3}3+{\log }_{3}5$$=1+{\log }_{3}5$

$\Rightarrow {\log }_{3}5=b-1$

${\log }_{30}18=\frac{{\log }_{3}18}{{\log }_{3}30}$$=\frac{{\log }_3(2.3^2)}{{\log }_3(2.3.5)}$

$=\frac{{\log }_{3}2+{\log }_3{3}^2}{{\log }_{3}2+{\log }_{3}3+{\log }_{3}5}$$=\frac{{\log }_{3}2+2}{{\log }_{3}2+1+{\log }_{3}5}$

$=\frac{\frac{1}{a}+2}{\frac{1}{a}+1+b-1}$$=\frac{2a+1}{ab+1}$

Vậy ${\log }_{30}18$$=\frac{2a+1}{ab+1}$