Question

If $2^a=3$ and $9^b=4$ then the value of $a.b$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (A)

$1$

$2^a=3$ $a{\log }_{10}2={\log }_{10}3$ $a=\frac{{\log }_{10}3}{{\log }_{10}2}$

$9^b=4$ $b{\log }_{10}9={\log }_{10}4$ $b=\frac{{\log }_{10}4}{{\log }_{10}9}$ $b=\frac{{\log }_{10}{2}^2}{{\log }_{10}{3}^3}$ $b=\frac{{\log }_{10}2}{{\log }_{10}3}$

$\therefore a.b=\frac{{\log }_{10}3}{{\log }_{10}2}\times \frac{{\log }_{10}2}{{\log }_{10}3}$

$\therefore a.b=1$