Basic Mathematics & Logarithm
Properties of Logarithms
Grade Class 11
Question:
<p>If \(a^{\log_3 7} = 27\), \(b^{\log_7 11} = 49\), and \(c^{\log_{11} 25} = \sqrt{11}\), then \(a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}\) equals</p>
\(489\)
\(469\)
\(464\)
\(400\)
Step-by-Step Solution
Key Concept: Raise each given relation once more by the same logarithmic exponent. From a^(log_3 7) = 27 = 3^3, we get a^((log_3 7)^2) = 7^3 = 343. Similarly b-term = 11^2 = 121 and c-term = 25^(1/2) = 5. Sum = 343 + 121 + 5 = 469.
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Raise each given relation once more by the same logarithmic exponent. From a^(log_3 7) = 27 = 3^3, we get a^((log_3 7)^2) = 7^3 = 343. Similarly b-term = 11^2 = 121 and c-term = 25^(1/2) = 5. Sum = 343 + 121 + 5 = 469. Trap: Use x^(log_a b) repeatedly instead of solving for a, b and c separately. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: B