If $a^x = b^y$, then

### Concept & Formula This problem requires converting an exponential equation into a logarithmic relationship. The key insight is taking the logarithm of both sides of an equation to bring variables out of the exponents. The primary rule used is the power rule of logarithms: $$ \log(m^n) = n \cdot \log(m) $$ ### Step-by-Step Solution **Given:** $a^x = b^y$ To isolate the exponents $x$ and $y$, we take the logarithm (to any common base, typically base 10) on both sides of the equation. Take log on both sides: $\log(a^x) = \log(b^y)$ Apply the power rule $\log(m^n) = n \cdot \log(m)$ to bring the exponents to the front: $x \cdot \log(a) = y \cdot \log(b)$ Now, we need to match the format of the options, which represent ratios. Let's isolate the ratio of the logarithms. Divide both sides by $x \cdot \log(b)$: $\frac{\log(a)}{\log(b)} = \frac{y}{x}$ ### Exam Strategy & Shortcut When equating exponents with different bases like $a^x = b^y$, immediately know that their logarithmic ratio is inversely proportional to their exponent ratio. Thus, $\log a$ over $\log b$ will always equal $y$ over $x$. Recognizing this pattern allows you to bypass the intermediate steps and directly select the correct ratio in seconds. ### Common Pitfall A very common mistake is confusing the inverse relationship when cross-multiplying, leading students to choose $\frac{\log a}{\log b} = \frac{x}{y}$. Always double-check your algebraic rearrangement: if $x$ is multiplied by $\log a$, then $x$ must end up in the denominator of the opposite side when setting up the fraction. ### Final Answer **Therefore, the correct answer is $\frac{\log a}{\log b} = \frac{y}{x}$.**