$\left[ \log \left( \frac{a^2}{bc} \right) + \log \left( \frac{b^2}{ac} \right) + \log \left( \frac{c^2}{ab} \right) \right]$ is equal to

### Concept & Formula This expression is simplified using the logarithmic product rule. The sum of multiple logarithms with the same base is equal to the logarithm of the product of their arguments. $$ \log x + \log y + \log z = \log(xyz) $$ ### Step-by-Step Solution * **Given:** A summation of three logarithmic terms containing algebraic fractions. $$ \log \left( \frac{a^2}{bc} \right) + \log \left( \frac{b^2}{ac} \right) + \log \left( \frac{c^2}{ab} \right) $$ * **Calculation:** Apply the product rule to merge the three logarithms into a single logarithmic expression. Multiply the three fractional arguments together. $$ \log \left( \frac{a^2}{bc} \times \frac{b^2}{ac} \times \frac{c^2}{ab} \right) $$ * Combine the numerators and denominators. $$ \log \left( \frac{a^2 \cdot b^2 \cdot c^2}{(bc) \cdot (ac) \cdot (ab)} \right) $$ * Rearrange and simplify the denominator by multiplying the common variables. $a \cdot a = a^2$, $b \cdot b = b^2$, and $c \cdot c = c^2$. $$ \log \left( \frac{a^2 b^2 c^2}{a^2 b^2 c^2} \right) $$ * The numerator and denominator are identical, so the fraction simplifies to 1. $$ \log(1) $$ * The logarithm of 1, regardless of the base, is always 0. $$ 0 $$ ### Exam Strategy & Shortcut When faced with cyclical symmetric expressions like $\frac{a^2}{bc}$, $\frac{b^2}{ac}$, and $\frac{c^2}{ab}$ being added together inside logs, you should suspect they will cancel out. A quick mental check multiplying the tops ($a^2b^2c^2$) and bottoms ($a^2b^2c^2$) confirms they reduce to 1. Since $\log(1) = 0$, you can select option (a) in under 10 seconds without writing anything down. ### Common Pitfall Some students attempt to apply the quotient rule to expand each term individually (e.g., $\log(a^2) - \log(bc)$). While mathematically correct, it creates a massive, sprawling expression that is highly prone to sign errors and takes significantly longer to resolve. Stick to the product rule for efficiency. ### Final Answer **Therefore, the correct answer is 0.**