<p>The number of real solutions of the equation \(\sqrt{\log_{10}(-x)} = \log_{10}(\sqrt{x^2})\) is</p>
Step-by-Step Solution
Key Concept: Since -x is inside the logarithm, x must be negative. For x < 0, sqrt(x^2) = -x. Let t = log_10(-x) with t >= 0. Then sqrt(t) = t, so t = 0 or 1. Hence -x = 1 or 10, giving x = -1 or -10. There are exactly two s...
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. Since -x is inside the logarithm, x must be negative. For x < 0, sqrt(x^2) = -x. Let t = log_10(-x) with t >= 0. Then sqrt(t) = t, so t = 0 or 1. Hence -x = 1 or 10, giving x = -1 or -10. There are exactly two solutions. Trap: Do not replace sqrt(x^2) by x; here x is forced to be negative. 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