Logarithm manipulation – simplify expression: Evaluate (log_a x)/(log_{ab} x) − log_a b.

Introduction / Context:
We simplify a compound expression using change-of-base and basic logarithm identities. The expression mixes bases a and ab.

Given Data / Assumptions:

• a > 0, a ≠ 1, x > 0, b > 0, b ≠ 1.
• Expression: (log_a x)/(log_{ab} x) − log_a b.

Concept / Approach:
Use change of base: log_{ab} x = (log_a x)/(log_a (ab)) = (log_a x)/(1 + log_a b). Then compute the ratio and subtract log_a b.

Step-by-Step Solution:
log_{ab} x = (log_a x)/(1 + log_a b).(log_a x)/(log_{ab} x) = (log_a x)/[(log_a x)/(1 + log_a b)] = 1 + log_a b.Therefore, (log_a x)/(log_{ab} x) − log_a b = (1 + log_a b) − log_a b = 1.

Verification / Alternative check:
Pick a = 10, b = 10, x arbitrary: then ab = 100 and the same simplification gives 1.

Why Other Options Are Wrong:
0 would require cancellation of the leading 1; a or ab are dimensionally inconsistent with a pure number result.

Common Pitfalls:
Misapplying change-of-base or forgetting that log_a a = 1.

Final Answer:
1