The value of $\left( \frac{1}{\log_3 60} + \frac{1}{\log_4 60} + \frac{1}{\log_5 60} \right)$ is

### Concept & Formula This problem is quickly solved by applying the **Base Change Formula** for logarithms, specifically the reciprocal property. When you take the reciprocal of a logarithm, the base and the argument swap places. $$ \frac{1}{\log_a b} = \log_b a $$ ### Step-by-Step Solution * **Given:** We need to evaluate the sum of three reciprocal logarithms. $$ \frac{1}{\log_3 60} + \frac{1}{\log_4 60} + \frac{1}{\log_5 60} $$ * **Calculation:** Apply the reciprocal rule to each term to unify the bases. Now, all logarithms will have a common base of 60. $$ \log_{60} 3 + \log_{60} 4 + \log_{60} 5 $$ * Use the product rule of logarithms, which states that the sum of logs with the same base equals the log of their product ($\log_b x + \log_b y = \log_b (xy)$). $$ \log_{60} (3 \times 4 \times 5) $$ * Multiply the numbers inside the argument. $$ \log_{60} (60) $$ * Apply the identity rule ($\log_a a = 1$). $$ \log_{60} 60 = 1 $$ ### Exam Strategy & Shortcut Whenever you see a sum of fractions in the format $\frac{1}{\log_x N} + \frac{1}{\log_y N} ...$, immediately recognize that it converts to $\log_N (x \times y ...)$. In this exact question, simply multiply the original bases $3 \times 4 \times 5 = 60$. Since the product equals the original argument $60$, the answer is instantly 1. ### Common Pitfall A major mistake is attempting to convert everything to base 10 using standard change-of-base ($\frac{\log 60}{\log 3}$) before taking the reciprocal. While mathematically sound, it creates complex fractions that consume valuable exam time and increase calculation errors. ### Final Answer **Therefore, the correct answer is 1.**