Question

Let $a=\log 3$ and $b=\frac{\log 3}{\log (\log 3)}($ All logarithms on base 10$)$. The number $a^b$ is

• A. an odd integer
• B. an even integer
• C. a prime number
• D. a composite

Answer 1

Answer: (A), (C)

Let ${\log }_{10}3=x$
$\therefore b=\frac{x}{\log x}\text{ and }a=x$
$\Rightarrow a^b=x^{\frac{x}{\log x}}=x^{x{\log }_{x}10}=x^{{\log }_{x}10^x}=10^x=10^{{\log }_{10}3}=3$