Telescoping product of logs: Evaluate log₂3 × log₃2 × log₃4 × log₄3.

Introduction / Context:
Products of logarithms with swapped bases often telescope. The identity log_b a × log_a b = 1 is the key.

Given Data / Assumptions:

• All logs are valid (bases and arguments positive and not equal to 1).

Concept / Approach:
Group the terms in pairs that invert bases and arguments: (log₂3 × log₃2) and (log₃4 × log₄3). Each pair equals 1.

Step-by-Step Solution:
log₂3 × log₃2 = 1.log₃4 × log₄3 = 1.Product = 1 × 1 = 1.

Verification / Alternative check:
Convert everything to natural logs: (ln 3/ln 2)(ln 2/ln 3)(ln 4/ln 3)(ln 3/ln 4) = 1.

Why Other Options Are Wrong:
Any value ≠ 1 would contradict the base-inversion identity.

Common Pitfalls:
Changing bases inconsistently or attempting unnecessary numeric approximations.

Final Answer:
1