If logₐ b = x (logarithm to base a), then what is the value of log_b a in terms of x?

Introduction / Context:
This problem checks conceptual understanding of logarithms, specifically the relationship between logₐ b and log_b a. It uses the basic definition of a logarithm and the idea that logarithms are inverses of exponentiation. The goal is to express log_b a in terms of x, where x is given as logₐ b. This forms a classic identity that is frequently used in simplifying logarithmic expressions and changing bases.

Given Data / Assumptions:

• logₐ b = x, that is, log base a of b equals x.
• Both a and b are positive and not equal to 1, which are standard conditions for logarithms.
• We need to find log_b a in terms of x.
• All logs have real values and are defined under usual logarithm rules.

Concept / Approach:
From the definition of logarithm, logₐ b = x means aˣ = b. We want log_b a, which by definition is the exponent to which b must be raised to produce a. Because a and b are related through aˣ = b, the relation between these logs becomes a simple reciprocal. Using either direct exponentiation or the change of base formula, we can show that log_b a is equal to 1 / x.

Step-by-Step Solution:
Step 1: Start with the given relation logₐ b = x. Step 2: By definition of logarithm, logₐ b = x implies aˣ = b. Step 3: We wish to find log_b a, which is the exponent y such that bʸ = a. Step 4: Substitute b = aˣ into bʸ = a to obtain (aˣ)ʸ = a. Step 5: Simplify the left side: (aˣ)ʸ = a^(x y). Step 6: Thus a^(x y) = a¹, which means x y = 1 because the bases are equal and positive. Step 7: Therefore y = 1 / x. But y is log_b a, so log_b a = 1 / x.

Verification / Alternative check:
An alternative method uses the change of base formula. We have logₐ b = (log b) / (log a) = x. Then log_b a = (log a) / (log b). Dividing the second by the first, log_b a = 1 / x, since (log a) / (log b) equals 1 divided by (log b / log a). Thus both methods confirm that log_b a is the reciprocal of logₐ b.

Why Other Options Are Wrong:
Expressions such as x / (x + 1), x / (1 − x), x / (x − 1), or x² do not arise from either the exponent definition or the change of base formula. If any of these were correct, substituting simple values like a = 2 and b = 8 would lead to contradictions. Only the reciprocal 1 / x is consistent with basic logarithm properties.

Common Pitfalls:
Many learners mistakenly think log_b a equals x or 1 minus x. Others try to manipulate the symbols without referring back to the clear meaning of logₐ b as an exponent. Remembering that logₐ b and log_b a are inverses in this specific way and checking with simple numerical choices for a and b is a good strategy to avoid errors.

Final Answer:
The required value is log_b a = 1 / x.