Question

If ${\log }_{10}x=y$, then the value of ${\log }_{1000}{x}^2$ in terms of $y$ is

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

Answer 1

The correct option is C $\frac{2y}{3}$

${\log }_{10}x=y$
$\Rightarrow \frac{2\log x}{3\log 10}=\frac{2y}{3}[\because {\log }_{b}a=\frac{\log a}{\log a}]$
$\Rightarrow \frac{\log x^2}{\log {10}^3}[\because x\log a=\log a^x]=\frac{2y}{3}$
$\Rightarrow {\log }_{1000}{x}^2=\frac{2y}{3}$