If $\log_{10} 125 + \log_{10} 8 = x$, then $x$ is equal to

### Concept & Formula This problem is a straightforward application of the product rule of logarithms. By combining the arguments, you create a power of the base ($10$), allowing for easy evaluation. Product Rule: $$ \log_a(M) + \log_a(N) = \log_a(M \cdot N) $$ ### Step-by-Step Solution **Given:** $\log_{10}(125) + \log_{10}(8) = x$ Since both logarithmic terms have the same base ($10$), use the product rule to combine them by multiplying their arguments: $\log_{10}(125 \cdot 8) = x$ Calculate the product inside the parenthesis: $125 \cdot 8 = 1000$ Substitute this back into the equation: $\log_{10}(1000) = x$ Now, express $1000$ as a power of the base $10$: $1000 = 10^3$ So, the equation becomes: $\log_{10}(10^3) = x$ Using the power rule ($\log_a(a^n) = n$), the logarithm of a base raised to a power evaluates to that power: $3 = x$ ### Exam Strategy & Shortcut Recognizing common number pairs that multiply to powers of $10$ is crucial for speed. $125 \times 8 = 1000$, $25 \times 4 = 100$, and $5 \times 2 = 10$. The moment you see $125$ and $8$ being added in base-10 logs, you should instantly know the argument becomes $1000$. Counting the three zeros in $1000$ gives you the answer $3$ immediately. ### Common Pitfall A common error is trying to evaluate $\log_{10}(125)$ and $\log_{10}(8)$ individually, which involves messy decimals or using log tables. This wastes time and introduces rounding errors. Always look to combine terms first to see if they simplify nicely. ### Final Answer **Therefore, the correct answer is 3.**