Evaluate the expression 1 / log₃ 60 + 1 / log₄ 60 + 1 / log₅ 60, where all logarithms are to their indicated bases.

Introduction / Context:
This question explores a useful identity involving reciprocals of logarithms with different bases but the same argument. The expression consists of three terms: 1 / log₃ 60, 1 / log₄ 60, and 1 / log₅ 60. By recognising how reciprocals of logs convert into logs with a common base, we can simplify the sum and show that it reduces to a very simple value.

Given Data / Assumptions:

• We need to evaluate S = 1 / log₃ 60 + 1 / log₄ 60 + 1 / log₅ 60.
• Here log₃ 60 means log of 60 to base 3, and similarly for bases 4 and 5.
• All logs are real and defined for positive arguments and bases not equal to 1.
• We may use the identity 1 / logₐ b = log_b a.

Concept / Approach:
The key identity is 1 / logₐ b = log_b a. This follows from the change of base formula. Applying this identity to each term in the sum will convert all the terms into logs with base 60. Specifically, 1 / log₃ 60 becomes log₆₀ 3, and similarly for the other terms. Then we can use properties of logarithms to combine the sum into a single log expression whose value is easy to recognise.

Step-by-Step Solution:
Step 1: Start with S = 1 / log₃ 60 + 1 / log₄ 60 + 1 / log₅ 60. Step 2: Use the identity 1 / logₐ b = log_b a for each term. Thus 1 / log₃ 60 = log₆₀ 3, 1 / log₄ 60 = log₆₀ 4, and 1 / log₅ 60 = log₆₀ 5. Step 3: Substitute to obtain S = log₆₀ 3 + log₆₀ 4 + log₆₀ 5. Step 4: Use the log property logₖ a + logₖ b = logₖ (a b) with common base 60. Combine the first two terms: log₆₀ 3 + log₆₀ 4 = log₆₀ (3 × 4) = log₆₀ 12. Step 5: Add the third term: log₆₀ 12 + log₆₀ 5 = log₆₀ (12 × 5) = log₆₀ 60. Step 6: Since log₆₀ 60 equals 1, the sum S equals 1.

Verification / Alternative check:
The identity 1 / logₐ b = log_b a can be re derived from the change of base formula. Using that, we have log₆₀ 3 + log₆₀ 4 + log₆₀ 5, which describes the log of the product 3 × 4 × 5 using base 60. The product is exactly 60, so log₆₀ 60 must be 1 by definition, confirming our result without needing numerical approximations.

Why Other Options Are Wrong:
Options 0, 5, and 60 are inconsistent with the identities used. There is no cancellation that would produce 0, and the product 3 × 4 × 5 is 60, not 1 or 60 squared. So the combined log cannot reasonably give 5 or 60. The sum simplifies precisely to log₆₀ 60, which is 1, making option 1 the only correct choice.

Common Pitfalls:
Students may try to compute each log numerically, which is unnecessary and error prone. Others might forget that the reciprocal identity changes both base and argument. Another frequent mistake is to combine logs with different bases directly, which is not allowed. The correct approach always converts them to a common base before combining.

Final Answer:
The value of 1 / log₃ 60 + 1 / log₄ 60 + 1 / log₅ 60 is 1.