If log base a of b plus log base b of a is equal to 2, that is log_a b + log_b a = 2, then which of the following relations between a and b must be true?

Introduction / Context:
This question checks understanding of logarithms with variable bases and arguments. In particular it tests the reciprocal relationship between log_a b and log_b a, and how to convert a logarithmic equation into a simpler algebraic equation. Such problems are very common in aptitude and algebra tests and they help students become comfortable working with logs whose base is not a fixed number like 10 or e.

Given Data / Assumptions:
- log_a b + log_b a = 2.
- a and b are positive real numbers and neither a nor b is equal to 1, so all logarithms are defined.
- We must find a relation between a and b that always holds under this condition.

Concept / Approach:
The key identity is that log_b a is the reciprocal of log_a b. More precisely, for positive a and b with a not equal to 1 and b not equal to 1, we have log_a b = 1 / log_b a. We use this to convert the given expression into a single variable equation. Once we reduce it to a standard quadratic equation, we solve it and interpret the result back in terms of a and b.

Step-by-Step Solution:
Let x = log_a b. Then, by the change of base property, log_b a = 1 / x. The given condition becomes x + 1 / x = 2. Multiply through by x to clear the denominator: x^2 + 1 = 2x. Rearrange to obtain a quadratic: x^2 - 2x + 1 = 0. Factor: (x - 1)^2 = 0, so x = 1. Thus log_a b = 1, which means b = a^1 = a. Therefore the relation that must hold is a = b.

Verification / Alternative check:
If a and b are equal, say a = b = k with k greater than 0 and not equal to 1, then log_a b = log_k k = 1 and log_b a = 1. Their sum is 1 + 1 = 2, which satisfies the given equation. If a is not equal to b, then log_a b is not 1, and the reciprocal sum x + 1 / x can never be equal to 2 unless x = 1. This confirms that a = b is the only possible relation.

Why Other Options Are Wrong:
a + b = 1: This is not forced by the logarithmic equation and may even fail for positive numbers greater than 1.
a - b = 1: This would only hold for very specific pairs, but the logarithmic equation demands a general relation that works for all valid a and b.
ab = 1: This also does not follow from the derived condition x = 1 and is unrelated to the sum of logs being 2.

Common Pitfalls:
One common mistake is forgetting that log_a b and log_b a are reciprocals, not equal values. Students might also incorrectly apply change of base and treat log_a b as log b / log a without keeping track of the base. Another error is failing to handle the quadratic equation correctly, for example by discarding the unique solution x = 1. Remembering that x + 1 / x has minimum value 2 when x is 1 also provides a useful shortcut check.

Final Answer:
The only relation that always holds under the given condition is a = b.