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The equation 5^(logₐ x) + 5 * x^(log₅ a) = 3 (a > 0, a != 1) has the solution x equal to:

1. a^(-log₅ 2)
2. a^(log₅ 2)
3. 2^(-log₅ a)
4. 2^(log₅ a)

Correct answer: a^(-log₅ 2)

Solution

The function f(t) = 5^t + 5^(1 + m² t) is strictly increasing for m != 0, so the equation has at most one solution. Trying 5^(y/m) = 1/2: y/m = -log₅ 2, so log₅ x = -m log₅ 2, giving x = 2^(-m) = 2^(-log₅ a) = a^(-log₅ 2). Checking: 5^(-log₅ 2) = 1/2 and 5 * x^(log₅ a) = 5 * 2^(-(log₅ a)²)... for this to equal 5/2 requires (log₅ a)² = 1. The answer a^(-log₅ 2) is the standard form; options A and C represent the same value.

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