Question

If $\frac{3}{2}\log a+\frac{2}{3}\log b-1=0$, then $a^9{b}^4$ is ${10}^m$. Then the value of $m$ is equal to

• A. $18$
• B. $12$
• C. $6$
• D. $13$

Answer 1

The correct option is C $6$

$\frac{3}{2}\log a+\frac{2}{3}\log b=1\text{Assuming base 10}\frac{3}{2}{\log }_{10}a+\frac{2}{3}{\log }_{10}b={\log }_{10}10{\log }_{10}{a}^{\frac{3}{2}}+{\log }_{10}{b}^{\frac{2}{3}}={\log }_{10}10{\log }_{10}{(a}^{\frac{3}{2}}{b}^{\frac{2}{3}})={\log }_{10}10\text{Removing log}{a}^{\frac{3}{2}}{b}^{\frac{2}{3}}=10\text{Cubing above}{a}^{\frac{9}{2}}{b}^2={10}^3\text{Squaring the above}{a}^9{b}^4={10}^6={10}^m$
So, $m=6$