Solve the simultaneous equations : \(\log_{2} x - \log_{2} y = 2 ; \log_{2} (x - 2y) = 3\)
Question 1 Report
Solve the simultaneous equations : \(\log_{2} x - \log_{2} y = 2 ; \log_{2} (x - 2y) = 3\)
Answer Details
From \(\log_2 x - \log_2 y = 2\):
\[\log_2\frac{x}{y} = 2 \;\Rightarrow\; \frac{x}{y} = 2^2 = 4 \;\Rightarrow\; x = 4y\]From \(\log_2(x - 2y) = 3\):
\[x - 2y = 2^3 = 8\]Substitute \(x = 4y\):
\[4y - 2y = 8 \;\Rightarrow\; 2y = 8 \;\Rightarrow\; y = 4,\quad x = 16\]\(x = 16,\ y = 4\). Check: \(\log_2 16 - \log_2 4 = 4 - 2 = 2\) and \(\log_2(16 - 8) = \log_2 8 = 3\).
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