If log(x − 5) = log(x) − log(5) and log(y − 6) = log(y) − log(6), compare x and y.

Introduction / Context:This problem tests log laws and equation solving. The identities transform into linear equations in x and y once we use log(a) − log(b) = log(a/b) and apply the one-to-one property of logarithms (same base, positive arguments).

Given Data / Assumptions:

• log(x − 5) = log(x) − log(5) ⇒ log(x − 5) = log(x/5).
• log(y − 6) = log(y) − log(6) ⇒ log(y − 6) = log(y/6).
• Log base is 10 (or any common base), arguments positive.

Concept / Approach:

• If log A = log B (same base), then A = B, provided A, B > 0.
• Solve the resulting linear equations to obtain x and y, then compare.

Step-by-Step Solution:

Verification / Alternative check:Both solutions satisfy domain constraints: x > 5 and y > 6 so all logs are defined, confirming validity.

Why Other Options Are Wrong:

• “x > y” and “x = y” contradict explicit solutions.
• “Can’t say” is wrong because the comparison is determined.

Common Pitfalls:

• Forgetting logs require positive arguments and using invalid x ≤ 5 or y ≤ 6.

Final Answer:x < y