Question

If $lo{g}_{10}x=y$, then which of the following options are correct?

• A. $lo{g}_{10}{x}^2=y^2$
• B. $lo{g}_{10}{x}^2=2y$
• C. $lo{g}_{1000}{x}^2=\frac{3y}{2}$
• D. $lo{g}_{1000}{x}^2=\frac{2y}{3}$

Answer 1

Answer: (B), (D)

The correct options are
B

$lo{g}_{10}{x}^2=2y$

D

$lo{g}_{1000}{x}^2=\frac{2y}{3}$

Given, $lo{g}_{10}x=y$
$\because lo{g}_{}{a}^m$ = m $lo{g}_{}a$
$\Rightarrow 2y=2lo{g}_{10}x$
$=lo{g}_{10}{x}^2$-------(1)
$\because lo{g}_{b^n}a$ = $\frac{1}{n}lo{g}_{b}a$
$\Rightarrow \frac{1}{3}\times 2y=\frac{1}{3}lo{g}_{10}{x}^2$
$\Rightarrow \frac{2y}{3}=lo{g}_{{10}^3}{x}^2$
$=lo{g}_{1000}{x}^2$------(2)
$\therefore$ Correct options are $lo{g}_{10}{x}^2=2y$ and $lo{g}_{1000}{x}^2=\frac{2y}{3}$