$(\log_5 3) \times (\log_3 625)$ equals

### Concept & Formula This problem is solved efficiently using the "Chain Rule" of logarithms, which is derived from the Change of Base formula. The Change of Base formula allows us to rewrite any logarithm as a fraction: $$ \log_a(b) = \frac{\log(b)}{\log(a)} $$ By multiplying two such logarithms, we get the Chain Rule property: $$ \log_a(b) \cdot \log_b(c) = \log_a(c) $$ ### Step-by-Step Solution **Given expression:** $(\log_5 3) \times (\log_3 625)$ Using the Change of Base formula, convert both logarithms to a common base (like base $10$): First term: $\log_5(3) = \frac{\log(3)}{\log(5)}$ Second term: $\log_3(625) = \frac{\log(625)}{\log(3)}$ Multiply the two fractions together: $\frac{\log(3)}{\log(5)} \times \frac{\log(625)}{\log(3)}$ Notice that $\log(3)$ appears in both the numerator and the denominator, so they cancel each other out completely: $\frac{\log(625)}{\log(5)}$ Now, convert this fraction back into a single logarithm using the reverse of the Change of Base formula: $\log_5(625)$ To evaluate this, express $625$ as a power of the base $5$: $625 = 5 \times 5 \times 5 \times 5 = 5^4$ Substitute this back: $\log_5(5^4)$ Using the power rule, this simplifies to just the exponent: $4$ ### Exam Strategy & Shortcut You do not need to convert to fractions on paper. The Chain Rule allows you to visually "cancel" the diagonal matching bases and arguments. In $(\log_5 3) \times (\log_3 625)$, the argument $3$ and base $3$ cancel instantly, leaving you with $\log_5(625)$. Since $5^4 = 625$, the answer is simply $4$. This takes $5$ seconds mentally. ### Common Pitfall A common mistake is attempting to multiply the arguments directly ($3 \times 625$) while keeping the bases separate, which violates all rules of logarithms. Logarithms cannot be multiplied internally like standard algebraic terms unless you are applying the change of base property. ### Final Answer **Therefore, the correct answer is 4.**