If log _{10} x + log _{10} y = 2,then the sma | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $\log _{10} x + \log _{10} y = 2$.Using the property $\log a + \log b = \log (ab)$,we get $\log _{10} (xy) = 2$.This implies $xy = 10^2 = 10

Vedclass · Accepted Answer

$20$

Vedclass · Answer

(C) Given $\log _{10} x + \log _{10} y = 2$.Using the property $\log a + \log b = \log (ab)$,we get $\log _{10} (xy) = 2$.This implies $xy = 10^2 = 100$.By the Arithmetic Mean-Geometric Mean $(AM-GM)$ inequality,for positive real numbers $x$ and $y$,$\frac{x+y}{2} \geq \sqrt{xy}$.Substituting $xy = 100$,we get $\frac{x+y}{2} \geq \sqrt{100}$.$\frac{x+y}{2} \geq 10$.Therefore,$x+y \geq 20$.The smallest possible value of $(x+y)$ is $20$.

If $\log _{10} x + \log _{10} y = 2$,then the smallest possible value of $(x + y)$ is

Similar Questions

The number of solutions of the equation $\log_{\sqrt{3}}(x^{3} - 1) = \log_{\sqrt{3}}(x - 1) + 2$ is:

If $p^3 = q^4 = r^6 = t^7 = s^2$,then $\log_t(pqrs) = \ldots$.

Suppose $\log _a b + \log _b a = c$. The smallest possible integer value of $c$ for all $a, b > 1$ is

If $x = \log_{3} 5$ and $y = \log_{17} 25$,which one of the following is correct?

If $a, b, c$ are distinct positive numbers,each different from $1$,such that $[(\log _b a)(\log _c a) - (\log _a a)] + [(\log _a b)(\log _c b) - (\log _b b)] + [(\log _a c)(\log _b c) - (\log _c c)] = 0$,then $abc =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module