Question

Let $x$ and $y$ be positive real numbers such that $\log_5(x + y) + \log_5(x - y) = 3$, and $\log_2 y - \log_2 x = 1 - \log_2 3$. Then $xy$ equals

Accepted Answer

B. First equation: $\log_5((x+y)(x-y)) = 3 \implies x^2 - y^2 = 5^3 = 125$. Second equation: $\log_2(y/x) = \log_2(2/3) \implies \frac{y}{x} = \frac{2}{3} \implies y = \frac{2}{3}x$. Substitute into first equation: $x^2 - \frac{4}{9}x^2 = 125 \implies \frac{5}{9}x^2 = 125 \implies x^2 = 225 \implies x = 15$. $y = \frac{2}{3}(15) = 10$. $xy = 15 	imes 10 = 150$.

System of Logarithmic Equations

Explanation

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