Question

If ${\log }_{b}n=2$ and ${\log }_{n}2b=2$ , then the value of $\log \left(b\right)$ is

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

Answer 1

The correct option is A $\frac{\log 2}{3}$

If ${\log }_{b}n=2$

Then $n=b^2$

Now, ${\log }_{n}2b=2$

$\Rightarrow {\log }_{b^2}2b=2$

$\Rightarrow \frac{\log \left(2b\right)}{\log \left(b^2\right)}=2$

$\Rightarrow \frac{\log 2+\log b}{2\log b}[\because \log x+\log y=\log (xy)\text{\&}\log a^x=x\log a]=2$

$\Rightarrow \log b=\frac{\log 2}{3}$