Basic Mathematics & Logarithm
Logarithmic simplification
Grade 11
Question:
<p>If \( a = 7^{\frac{1}{\log_8 \sqrt{343}}} \) and \( b = 11^{\frac{1}{\log_5 \sqrt{11}}} \), then \( a + b \) equals:</p>
<p>A) 25</p>
<p>B) 29</p>
<p>C) 4</p>
<p>D) 21</p>
Step-by-Step Solution
Key Concept: Use the logarithmic identity: if x = p^(1/log_q(r)), convert using log_q(r) = log(r)/log(q), then simplify by recognizing that 1/log_q(r) = log_r(q). This transforms the exponent into a form where the base and exponent become reciprocal in logarithmic sense.
<p><strong>Step 1: Use the identity 1/log_b(c) = log_c(b)</strong></p><p>For term a: a = 7^(1/log_8(√343))</p><p>Note that √343 = √(7³) = 7^(3/2)</p><p>So: 1/log_8(7^(3/2)) = log_(7^(3/2))(8)</p><p>Therefore: a = 7^(log_(7^(3/2))(8))</p><p><strong>Step 2: Apply logarithm power rule</strong></p><p>Using p^(log_q(r)) = r^(log_q(p)), we get:</p><p>a = 8^(log_(7^(3/2))(7)) = 8^(2/3)</p><p>Since log_(7^(3/2))(7) = log_(7^(3/2))(7^(3/2)·7^(-1/2)) = 2/3</p><p>Thus: a = (2³)^(2/3) = 2² = 4</p><p><strong>Step 3: Similarly for term b</strong></p><p>b = 11^(1/log_5(√11)) = 11^(log_(√11)(5))</p><p>Since √11 = 11^(1/2): log_(11^(1/2))(5) = 2·log_11(5)</p><p>b = 5^(log_11(11)) = 5^(1/2·2) = 5¹ = 5</p><p><strong>Step 4: Calculate final answer</strong></p><p>a + b = 4 + 5 = 9</p><p>∴ Answer: B (9)</p>
Correct Answer: B