If $\log_{10} 7 = a$, then $\log_{10} \left( \frac{1}{70} \right)$ is equal to

### Concept & Formula This problem requires applying the quotient rule and product rule of logarithms to break down the given expression, alongside the fundamental identity that the logarithm of the base itself is 1. $$ \log \left(\frac{x}{y}\right) = \log x - \log y $$ $$ \log(xy) = \log x + \log y $$ ### Step-by-Step Solution * **Given:** We know that $\log_{10} 7 = a$. We need to evaluate $\log_{10} \left( \frac{1}{70} \right)$. * **Calculation:** First, apply the quotient rule to expand the target expression. $$ \log_{10} \left( \frac{1}{70} \right) = \log_{10} 1 - \log_{10} 70 $$ * The logarithm of 1 to any base is always 0 ($\log_{10} 1 = 0$). $$ 0 - \log_{10} 70 = -\log_{10} 70 $$ * Now, break down 70 into factors that include the given value (7) and the base (10), which is $7 \times 10$. Apply the product rule. $$ - (\log_{10} (7 \times 10)) $$ $$ - (\log_{10} 7 + \log_{10} 10) $$ * Substitute the given value $\log_{10} 7 = a$, and use the identity $\log_{10} 10 = 1$. $$ - (a + 1) $$ ### Exam Strategy & Shortcut Instead of expanding via the quotient rule first, you can use negative exponents: $\frac{1}{70} = 70^{-1}$. Then, $\log_{10}(70^{-1}) = -\log_{10} 70$. Split 70 into $7 \times 10$. $-\log_{10}(7 \times 10) = -(\log_{10} 7 + \log_{10} 10) = -(a + 1)$. This gets you to the answer in fewer mental steps. ### Common Pitfall Students often incorrectly handle the negative sign when expanding $- \log_{10}(7 \times 10)$, forgetting to distribute it to both terms, which leads to the wrong answer $-a + 1$ instead of $-(a + 1)$. Always use parentheses when applying logarithmic rules after a negative sign. ### Final Answer **Therefore, the correct answer is $-(1 + a)$.**