The value of $\left[ \frac{1}{\log_{(p/q)} x} + \frac{1}{\log_{(q/r)} x} + \frac{1}{\log_{(r/p)} x} \right]$ is

### Concept & Formula This question tests your ability to use the base change reciprocal property to convert fractions into manageable additive logarithms. $$ \frac{1}{\log_b a} = \log_a b $$ $$ \log_x a + \log_x b = \log_x(ab) $$ ### Step-by-Step Solution * **Given:** An expression with the variable $x$ as the argument in the denominators, and fractions as the bases. $$ \frac{1}{\log_{(p/q)} x} + \frac{1}{\log_{(q/r)} x} + \frac{1}{\log_{(r/p)} x} $$ * **Calculation:** Use the reciprocal rule to bring each logarithm up to the numerator. This swaps the fractional base with the argument $x$, giving all three logarithms a common base of $x$. $$ \log_x \left(\frac{p}{q}\right) + \log_x \left(\frac{q}{r}\right) + \log_x \left(\frac{r}{p}\right) $$ * Since all logarithms now share the same base $x$ and are being added together, apply the product rule to combine them. Multiply the arguments together inside a single logarithm. $$ \log_x \left( \frac{p}{q} \times \frac{q}{r} \times \frac{r}{p} \right) $$ * Simplify the product inside the argument. The $p$ in the numerator cancels the $p$ in the denominator, the $q$ cancels the $q$, and the $r$ cancels the $r$. $$ \log_x (1) $$ * The logarithm of $1$ to any valid base is exactly $0$. $$ 0 $$ ### Exam Strategy & Shortcut Identify the common structure: you have reciprocal logs that all share the same argument $x$. This means they will all instantly convert to logs with base $x$. Because it's an addition chain, you just need to multiply the original bases together. Visually cross-canceling $(p/q) \times (q/r) \times (r/p)$ yields $1$ in your head. Since $\log_x(1) = 0$, you can mark the answer in under 10 seconds. ### Common Pitfall A major misstep is trying to use the quotient rule to expand the bases in the denominator (e.g., trying to turn $\log_{(p/q)} x$ into a subtraction). This creates complex nested fractions that are incredibly difficult to resolve. Always look to unify the bases first, usually via the reciprocal rule. ### Final Answer **Therefore, the correct answer is 0.**