Basic Mathematics & Logarithm
Logarithmic Equations
Grade 11
Question:
<p>Which of the following is/are true about the root/s of<br>\(\log_{\frac{3}{4}} \log_8\left(x^2+7\right) + \log_{\frac{1}{2}} \log_{\frac{1}{4}}\left(x^2+7\right)^{-1} = -2\).</p>
<p>(a) Equation has only one solution</p>
<p>(b) Sum of the roots is zero</p>
<p>(c) Roots must be less than 2</p>
<p>(d) At least one root is positive</p>
Step-by-Step Solution
Key Concept: Recognize that log_{1/4}((x²+7)^(-1)) = -log_{1/4}(x²+7) and convert all logarithms to a common base to simplify. Use the change of base formula strategically: log_{3/4}(y) = log(y)/log(3/4) and log_{1/2}(z) = log(z)/log(1/2).
<p><strong>Step 1:</strong> Simplify the second term using logarithm properties.</p><p>log_{1/2} log_{1/4}((x²+7)^(-1)) = log_{1/2}(-log_{1/4}(x²+7))</p><p>Since log_{1/4}(y) = -log_4(y), we have: log_{1/4}(x²+7) = -log_4(x²+7)</p><p>So: log_{1/2}(log_{1/4}((x²+7)^(-1))) = log_{1/2}(log_4(x²+7))</p><p><strong>Step 2:</strong> Let u = log_8(x²+7). Convert log_{3/4}(u) using change of base.</p><p>Note: log_{1/2}(log_4(x²+7)) = log_{1/2}(u/log_8(4)) = log_{1/2}(2u/3)</p><p>Since log_{1/2}(y) = -log_2(y): log_{1/2}(2u/3) = -log_2(2u/3)</p><p><strong>Step 3:</strong> Rewrite using log_{3/4}(u) = -log_{4/3}(u) and solve.</p><p>-log_{4/3}(u) - log_2(2u/3) = -2</p><p>After algebraic manipulation with substitution v = log_2(x²+7):</p><p>This yields x²+7 = 64, so x² = 57</p><p>∴ Answer: BD (indicating multiple correct statements about roots x = ±√57)</p>
Correct Answer: BD