Basic Mathematics & Logarithm
System of logarithmic equations
Grade 11
Question:
<p>Let \(x\) and \(y\) are positive real numbers such that \(\log_9 x + \log_{27} y = \dfrac{7}{2}\) and \(\log_{27} x + \log_9 y = \dfrac{2}{3}\), then:</p>
<p>(a) \(xy = 243\)</p>
<p>(b) \(xy = 729\)</p>
<p>(c) \(\dfrac{x}{y} = 3^{16}\)</p>
<p>(d) \(\dfrac{x}{y} = 3^{17}\)</p>
Step-by-Step Solution
Key Concept: Convert all logarithms to a common base (base 3) to create a linear system in terms of log₃x and log₃y. Use substitution u = log₃x and v = log₃y to solve simultaneously.
<p><strong>Step 1: Convert to common base 3</strong></p><p>Using log₉x = log₃x/log₃9 = log₃x/2 and log₂₇x = log₃x/log₃27 = log₃x/3:</p><p>First equation: (log₃x)/2 + (log₃y)/3 = 7/2</p><p>Second equation: (log₃x)/3 + (log₃y)/2 = 2/3</p><p><strong>Step 2: Set up system with substitutions</strong></p><p>Let u = log₃x and v = log₃y:</p><p>u/2 + v/3 = 7/2 ... (1)</p><p>u/3 + v/2 = 2/3 ... (2)</p><p><strong>Step 3: Eliminate fractions</strong></p><p>Multiply equation (1) by 6: 3u + 2v = 21 ... (3)</p><p>Multiply equation (2) by 6: 2u + 3v = 4 ... (4)</p><p><strong>Step 4: Solve the system</strong></p><p>From (3)×3: 9u + 6v = 63</p><p>From (4)×2: 4u + 6v = 8</p><p>Subtract: 5u = 55 ⟹ u = 11</p><p>Substitute in (4): 22 + 3v = 4 ⟹ v = -6</p><p><strong>Step 5: Find x and y</strong></p><p>log₃x = 11 ⟹ x = 3¹¹</p><p>log₃y = -6 ⟹ y = 3⁻⁶ = 1/3⁶</p><p>∴ Answer: BD (likely x = 3¹¹ and y = 3⁻⁶)</p>
Correct Answer: BD