Question:

<p><strong>Step 1: Convert the first logarithmic equation</strong></p><p>Given: $\log_{10}(2x) + \log_{10} y = 2$</p><p>Using $\log a + \log b = \log(ab)$:</p><p>$\log_{10}(2xy) = 2$</p><p>Therefore: $2xy = 10^2 = 100$</p><p>So: $xy = 50$ ... (Equation 1)</p><p><strong>Step 2: Convert the second logarithmic equation</strong></p><p>Given: $\log_{10} x^2 - \log_{10}(2y) = 4$</p><p>Using $\log a - \log b = \log(a/b)$:</p><p>$\log_{10}\left(\frac{x^2}{2y}\right) = 4$</p><p>Therefore: $\frac{x^2}{2y} = 10^4 = 10000$</p><p>So: $x^2 = 20000y$ ... (Equation 2)</p><p><strong>Step 3: Substitute Equation 1 into Equation 2</strong></p><p>From Equation 1: $y = \frac{50}{x}$</p><p>Substituting into Equation 2:</p><p>$x^2 = 20000 \cdot \frac{50}{x}$</p><p>$x^3 = 1000000$</p><p>$x = 100$</p><p><strong>Step 4: Find y</strong></p><p>From Equation 1: $xy = 50$</p><p>$y = \frac{50}{100} = \frac{1}{2}$</p><p><strong>Step 5: Verify the solution</strong></p><p>Check Equation 1: $100 \cdot \frac{1}{2} = 50$ ✓</p><p>Check Equation 2: $\frac{100^2}{2 \cdot \frac{1}{2}} = \frac{10000}{1} = 10000$ ✓</p><p><strong>Step 6: Calculate x + y and find m and n</strong></p><p>$x + y = 100 + \frac{1}{2} = \frac{201}{2}$</p><p>Therefore: $m = 201$, $n = 2$ (which are coprime)</p><p><strong>Step 7: Calculate m - 3n⁶</strong></p><p>$m - 3n^6 = 201 - 3(2)^6 = 201 - 3(64) = 201 - 192 = 9$</p><p><strong>∴ Answer: 9</strong></p>