Question

If ${\log }_{a}3=2$ and ${\log }_{b}8=3$, then ${\log }_{a}b$ wiil be

• A. ${\log }_{3}2$
• B. ${\log }_{2}3$
• C. ${\log }_{3}4$
• D. ${\log }_{4}3$

Answer 1

The correct option is C ${\log }_{3}4$

${\log }_{a}3=2$

$\Rightarrow \frac{\log 3}{\log a}[\because {\log }_{b}a=\frac{\log a}{\log b}]=2$

$\Rightarrow \log a=\frac{\log 3}{2}$ .............(1)
${\log }_{b}8=3$

$\Rightarrow \frac{\log 8}{\log b}=3$

$\Rightarrow \log b=\frac{\log 8}{3}=\frac{\log 2^3}{3}=\log 2[\because \log a^m=m\log a]$ .....................(2)

${\log }_{a}b=\frac{\log b}{\log a}=\frac{2\log 2}{\log 3}=\frac{\log 4}{\log 3}$ ...{ from (1) & (2)}

$\Rightarrow {\log }_{a}b={\log }_{3}4$
Ans:C