Question

The value of $a^{\frac{{\log }_b\left({\log }_{b}N\right)}{{\log }_{b}a}}$ is

• A. ${\log }_b(N-b)$
• B. ${\log }_b(N+b)$
• C. ${\log }_b\frac{N}{b}$
• D. ${\log }_{b}N$

Answer 1

Answer: (D)

${\log }_{b}N$

Using ${\log }_{p}q=\frac{\log q}{\log p}$

$=a^{\frac{{\log }_b\left({\log }_{b}N\right)}{{\log }_{b}a}}$

$=a^{\frac{\log \left({\log }_{b}N\right)/\log b}{\log a/\log b}}$
$=a^{{\log }_a\left({\log }_{b}N\right)}$
$={\log }_{b}N$ $[\because a^{{\log }_{a}b}=b]$