$\frac{1}{\log_a b} \times \frac{1}{\log_b c} \times \frac{1}{\log_c a}$ is equal to

### Concept & Formula This problem requires two core properties of logarithms: the Reciprocal Rule (a derivation of the base-change formula) and the Chain Rule. $$ \frac{1}{\log_x y} = \log_y x $$ $$ \log_x y \times \log_y z = \log_x z $$ ### Step-by-Step Solution * **Given:** A product of three reciprocal logarithms with varying bases. $$ \frac{1}{\log_a b} \times \frac{1}{\log_b c} \times \frac{1}{\log_c a} $$ * **Calculation:** First, apply the reciprocal rule to move the logarithms from the denominator to the numerator. This swaps the base and the argument for each term. $$ \log_b a \times \log_c b \times \log_a c $$ * Rearrange the terms to make the chain rule visually obvious. Place them so the argument of one log matches the base of the next. $$ \log_c b \times \log_b a \times \log_a c $$ * Apply the logarithmic chain rule to the first two terms ($\log_c b \times \log_b a$). The $b$'s cancel out, collapsing the terms. $$ \log_c a \times \log_a c $$ * Apply the chain rule again to the remaining terms. The $a$'s cancel out. $$ \log_c c $$ * Apply the identity rule for logarithms, where a base to the power of 1 equals itself ($\log_x x = 1$). $$ 1 $$ ### Exam Strategy & Shortcut An alternative and often faster visualization uses the standard Base Change Formula: $\log_x y = \frac{\log y}{\log x}$. Applying this to the reciprocal format means $\frac{1}{\log_a b} = \frac{\log a}{\log b}$. Transforming all three terms gives: $\frac{\log a}{\log b} \times \frac{\log b}{\log c} \times \frac{\log c}{\log a}$. You can immediately see that every numerator perfectly cancels every denominator, leaving exactly 1. ### Common Pitfall A major trap is confusing multiplication with addition. Students sometimes mistakenly try to convert the product of these fractions into the logarithm of a sum, which is mathematically impossible. Always rely on the base change formula when multiplying logarithms with different bases. ### Final Answer **Therefore, the correct answer is 1.**