Find the value of log base 5 of 5 multiplied by log base 4 of 9 multiplied by log base 3 of 2, that is log_5 5 × log_4 9 × log_3 2.

Introduction / Context:
This question tests the understanding of how logarithms with different bases can simplify when they are multiplied together. The expression involves three logs with bases 5, 4 and 3 respectively, and we must evaluate their product. Recognising which parts simplify to 1 and how to use the change of base formula is the key to solving this quickly in an exam.

Given Data / Assumptions:
- The expression to evaluate is E = log_5 5 × log_4 9 × log_3 2.
- The bases 5, 4 and 3 are positive numbers not equal to 1, so all logs are defined.
- We are looking for a numerical value, not a symbolic expression.

Concept / Approach:
First, note that log_5 5 is equal to 1 because any log of the same base and argument equals 1. Next, we relate log_4 9 and log_3 2. By writing them in terms of a common base using the change of base formula, we can see that log_4 9 and log_3 2 are reciprocals. The product of a number and its reciprocal is 1, and since the first factor is already 1, the entire expression simplifies to 1.

Step-by-Step Solution:
Start with E = log_5 5 × log_4 9 × log_3 2. Observe that log_5 5 = 1 because 5^1 = 5. So E = 1 × log_4 9 × log_3 2 = log_4 9 × log_3 2. Use change of base for log_4 9: log_4 9 = ln 9 / ln 4. Write 9 and 4 as powers: 9 = 3^2 and 4 = 2^2. Then ln 9 = 2 ln 3 and ln 4 = 2 ln 2, so log_4 9 = (2 ln 3) / (2 ln 2) = ln 3 / ln 2. Use change of base for log_3 2: log_3 2 = ln 2 / ln 3. Now multiply: log_4 9 × log_3 2 = (ln 3 / ln 2) × (ln 2 / ln 3). The ln 3 and ln 2 terms cancel, leaving 1. Therefore E = 1.

Verification / Alternative check:
We can numerically approximate the values to confirm. log_5 5 is exactly 1. log_4 9 can be approximated as ln 9 / ln 4, and log_3 2 as ln 2 / ln 3. Multiplying these two approximations gives a result extremely close to 1.0, confirming the algebraic simplification. This shows that both the symbolic reasoning and numerical approximation agree.

Why Other Options Are Wrong:
3/2: This would require log_4 9 × log_3 2 to be 1.5, which contradicts the reciprocal relationship between log_4 9 and log_3 2.
2: This would occur only if log_4 9 and log_3 2 were both equal to the same number greater than 1, which is not true.
5: There is no factor of 5 that remains after simplification, so this is unrelated to the correct product.

Common Pitfalls:
Students sometimes forget that log_a a is 1 and treat it as an unknown. Another common mistake is to misapply the change of base formula, for example writing log_4 9 as ln 4 / ln 9. This reverses the ratio and destroys the reciprocal relationship with log_3 2. Some may also try to approximate each log numerically without simplification, which is slower and prone to rounding error. Systematically using change of base with the correct numerator and denominator avoids these issues.

Final Answer:
The value of log_5 5 × log_4 9 × log_3 2 is 1.