Solve the simultaneous equations : \(\log_{2} x - \log_{2} y = 2 ; \log_{2} (x - 2y) = 3\)
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Correct Answer: Option n
Explanation:
\(\log_{2} x - \log_{2} y = 2 \implies \log_{2} (\frac{x}{y}) = 2 \)
\(\frac{x}{y} = 2^{2} = 4 \implies x = 4y ... (1)\)
\(\log_{2} (x - 2y) = 3 \implies x - 2y = 2^{3} = 8 ... (2)\)
Putting (1) into (2),
\(4y - 2y = 8 \implies 2y = 8\)
\(y = 4\)
\(x = 4y \implies x = 4(4) = 16\)
\(x = 16 ; y = 4\)
\(\log_{2} x - \log_{2} y = 2 \implies \log_{2} (\frac{x}{y}) = 2 \)
\(\frac{x}{y} = 2^{2} = 4 \implies x = 4y ... (1)\)
\(\log_{2} (x - 2y) = 3 \implies x - 2y = 2^{3} = 8 ... (2)\)
Putting (1) into (2),
\(4y - 2y = 8 \implies 2y = 8\)
\(y = 4\)
\(x = 4y \implies x = 4(4) = 16\)
\(x = 16 ; y = 4\)