If logₐ (a b) = x, where a and b are positive and a ≠ 1, then what is the value of log_b (a b) in terms of x?

Introduction / Context:
This is a conceptual logarithm question that asks you to relate two different logarithmic expressions involving the same product a b but different bases. It tests your understanding of the change of base idea and the relationship between a logarithm and its reciprocal when bases are swapped.

Given Data / Assumptions:

• logₐ (a b) = x.
• a and b are positive real numbers with a ≠ 1 and b ≠ 1.
• We must find log_b (a b) expressed in terms of x.

Concept / Approach:
From the equation logₐ (a b) = x we can convert to exponential form and express b in terms of a and x. Then we find log_b (a b) using the definition of logarithms and the property that log_b a is the reciprocal of logₐ b. Combining these relationships allows us to express the required logarithm purely in terms of x.

Step-by-Step Solution:
Start with logₐ (a b) = x. By definition, this means a^x = a b. Divide both sides by a: b = a^(x - 1). We need log_b (a b). By logarithm rules, log_b (a b) = log_b a + log_b b. Note that log_b b = 1. Next, log_b a = 1 / logₐ b (since logs with swapped bases are reciprocals). From b = a^(x - 1), we have logₐ b = x - 1. Thus log_b a = 1 / (x - 1). Therefore, log_b (a b) = log_b a + 1 = 1 / (x - 1) + 1. Combine the terms: 1 / (x - 1) + 1 = (1 + x - 1) / (x - 1) = x / (x - 1).

Verification / Alternative check:
We can test with a simple numerical example. Let a = 2 and let b = 4. Then a b = 8. Compute log₂ 8 = 3, so x = 3. According to our formula, log_b (a b) = x / (x - 1) = 3 / 2. Indeed, log₄ 8 = log 8 / log 4 = 3 log 2 / (2 log 2) = 3 / 2, which matches perfectly.

Why Other Options Are Wrong:
1/x: This ignores the effect of the added factor a inside the logarithm and does not match the test example.
x/(x+1) and x/(1-x): Both give incorrect values when tested with simple numerical choices of a and b, and they do not follow from the algebraic steps above.

Common Pitfalls:
Learners often forget to convert the initial logarithmic equation into exponential form or mishandle the reciprocal relationship between logₐ b and log_b a. Missing the step log_b (a b) = log_b a + 1 is another common error, as is algebraic slippage when simplifying 1 / (x - 1) + 1.

Final Answer:
The required value is x / (x - 1).