Question

If ${\log }_{a}3=2$ and ${\log }_{b}8=3$ then ${\log }_{a}b$ is -

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

Answer 1

Answer: (C)

${\log }_{3}4$

Given,

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

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

And,

${\log }_{b}8=3$

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

Then,

${\log }_{a}b=\frac{\log b}{\log a}=\frac{\log 8}{3}\times \frac{2}{\log 3}=\frac{3\log 2\times 2}{3\log 3}$

$\Rightarrow {\log }_{a}b=\frac{\log 4}{\log 3}={\log }_{3}4$

So, option $C$ is correct