<p>The sum of all solutions of the equation \(\log_{10} x + \log_{10}(x + 2) - \log_{10}(5x + 4) = 0\) is</p>
Step-by-Step Solution
Key Concept: Combine logarithms and solve the resulting quadratic under the domain restriction. The equation becomes log_10[x(x + 2)/(5x + 4)] = 0, so x(x + 2) = 5x + 4. Thus x^2 - 3x - 4 = 0, giving x = 4 or -1. Domain requires x...
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Combine logarithms and solve the resulting quadratic under the domain restriction. The equation becomes log_10[x(x + 2)/(5x + 4)] = 0, so x(x + 2) = 5x + 4. Thus x^2 - 3x - 4 = 0, giving x = 4 or -1. Domain requires x > 0, so only x = 4 survives. The sum is 4. Trap: Always check domain after solving the algebraic equation. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: C