If 5x=(0.5)y=1000, then 1x−1y=

5x=(0.5)y=10005x=103⇒5=103x→(1)(0.5)y=103⇒0.5=03y⇒5=103y+1→(2)From (1) and (2), 103x=103y+13x−3y=1⇒1x−1y=13

If 5x=(0.5)y=1000, then 1x−1y=

5x=(0.5)y=10005x=103⇒5=103x→(1)(0.5)y=103⇒0.5=03y⇒5=103y+1→(2)From (1) and (2), 103x=103y+13x−3y=1⇒1x−1y=13

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If $5^{x} = {(0.5)}^{y} = 1000$ , then $\frac{1}{x} - \frac{1}{y} =$

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$\frac{1}{2}$

$\frac{1}{3}$

$\frac{1}{4}$

answer is C.

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Detailed Solution

$5^{x} = {(0.5)}^{y} = 1000$

$5^{x} = 10^{3} \Rightarrow 5 = 10^{\frac{3}{x}} \to (1)$

${(0.5)}^{y} = 10^{3} \Rightarrow 0.5 = 0^{\frac{3}{y}} \Rightarrow 5 = 10^{\frac{3}{y} + 1} \to (2)$

From (1) and (2), $10^{\frac{3}{x}} = 10^{\frac{3}{y} + 1}$

$\frac{3}{x} - \frac{3}{y} = 1 \Rightarrow \frac{1}{x} - \frac{1}{y} = \frac{1}{3}$

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If $5^{x} = {(0.5)}^{y} = 1000$ , then $\frac{1}{x} - \frac{1}{y} =$