Question

If $lo{g}_{10}x+lo{g}_{10}y\ge 2$, then the smallest value of $x+y$ is?

• A. $10$
• B. $30$
• C. $20$
• D. None of these

Answer 1

The correct option is C $20$

$lo{g}_{10}x+lo{g}_{10}y\ge 2$

$lo{g}_{10}xy\ge 2$ [since $loga+logb=log\left(ab\right)$)

$⟹xy\ge {10}^2=100$

Here we also have

$x>0$ and $y>0$ since $x$ and $y$ are under $log$

And we know $AM\ge GM$

$\frac{x+y}{2}\ge \sqrt{xy}$

$\frac{x+y}{2}\ge \sqrt{100}=10$

$x+y\ge 20$