If $\log_{10} x + \log_{10} 5 = 2$, then $x$ equals

### Concept & Formula This problem applies the product rule of logarithms to combine terms, followed by converting the logarithmic equation into its exponential form. Product Rule: $$ \log_a(M) + \log_a(N) = \log_a(M \cdot N) $$ ### Step-by-Step Solution **Given:** $\log_{10}(x) + \log_{10}(5) = 2$ Use the product rule to combine the two logarithmic terms on the left side of the equation. Since they have the same base ($10$), we multiply their arguments: $\log_{10}(x \cdot 5) = 2$ $\log_{10}(5x) = 2$ Now, convert the logarithmic equation to its exponential form to solve for $x$: $10^2 = 5x$ Simplify the exponent: $100 = 5x$ Divide both sides by $5$ to isolate $x$: $x = \frac{100}{5}$ $x = 20$ ### Exam Strategy & Shortcut Instead of combining the logs, you can recognize that $2$ is $\log_{10}(100)$. So, the equation becomes $\log_{10}(x) + \log_{10}(5) = \log_{10}(100)$. Using the product rule visually, you know $x \times 5$ must equal $100$. What times $5$ is $100$? $20$. This mental calculation avoids writing out the exponential conversion steps entirely. ### Common Pitfall A frequent beginner mistake is attempting to add the arguments directly, interpreting $\log_{10}(x) + \log_{10}(5)$ as $\log_{10}(x + 5)$. Remember that addition on the outside of a logarithm translates to multiplication on the inside. ### Final Answer **Therefore, the correct answer is 20.**