<p>The solution of the equation \(5^{\log_a x} + 5x^{\log_a 5} = 3\), where \(a>0\) and \(a \ne 1\), is</p>
Step-by-Step Solution
Key Concept: Use x^(log_a 5) = 5^(log_a x). Let y = 5^(log_a x). Then the equation becomes y + 5y = 3, which reduces to y = 1/2 after using the common transformation. Solving for x gives x = a^(-log_5 2), which is equivalent to 2^...
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. Use x^(log_a 5) = 5^(log_a x). Let y = 5^(log_a x). Then the equation becomes y + 5y = 3, which reduces to y = 1/2 after using the common transformation. Solving for x gives x = a^(-log_5 2), which is equivalent to 2^(-log_5 a). Trap: Recognize the symmetry x^(log_a 5) = 5^(log_a x); otherwise the equation looks nonlinear in two different ways. 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: A, C