If $\log_{10} 5 + \log_{10} (5x + 1) = \log_{10} (x + 5) + 1$, then $x$ is equal to

### Concept & Formula This equation requires condensing multiple logarithmic terms into a single term on both sides using the product rule. A key step is recognizing that an isolated integer must be converted into a logarithm before the equation can be resolved. Product Rule: $$ \log_a(M) + \log_a(N) = \log_a(M \cdot N) $$ Constant Conversion: $$ 1 = \log_{10}(10) $$ ### Step-by-Step Solution **Given:** $\log_{10}(5) + \log_{10}(5x + 1) = \log_{10}(x + 5) + 1$ **Step 1: Simplify the Left Side** Apply the product rule to the left side, as both terms share base $10$: $\log_{10}(5 \cdot (5x + 1))$ $\log_{10}(25x + 5)$ **Step 2: Simplify the Right Side** To combine the terms on the right, we must express the integer $1$ as a base-$10$ logarithm. We know that $\log_{10}(10) = 1$. Rewrite the right side: $\log_{10}(x + 5) + \log_{10}(10)$ Now apply the product rule: $\log_{10}(10 \cdot (x + 5))$ $\log_{10}(10x + 50)$ **Step 3: Equate the Arguments** Our simplified equation is now: $\log_{10}(25x + 5) = \log_{10}(10x + 50)$ Since the logarithmic bases on both sides are identical, their arguments must be equal: $25x + 5 = 10x + 50$ **Step 4: Solve the Linear Equation** Subtract $10x$ from both sides: $15x + 5 = 50$ Subtract $5$ from both sides: $15x = 45$ Divide by $15$: $x = 3$ ### Exam Strategy & Shortcut Rather than solving the algebra, plug the options into the equation starting from the smallest integer. Try $x = 3$: Left Side: $\log(5) + \log(16) = \log(80)$ Right Side: $\log(8) + 1 = \log(8) + \log(10) = \log(80)$ The left and right sides match instantly. Option checking on linear log equations takes a fraction of the time compared to algebraic condensing. ### Common Pitfall The most frequent mistake is ignoring the constant $1$ on the right side of the equation and incorrectly equating $(25x + 5)$ directly to $(x + 5)$. Remember that every single term must be incorporated into the logarithm before you can drop the "log" notation from both sides. ### Final Answer **Therefore, the correct answer is 3.**