<p>The product of all solutions of the equation \(x^{1 + \log_{10} x} = 100000x\) is</p>
Step-by-Step Solution
Key Concept: Let t = log_10 x so x = 10^t. Then x^(log_10 x) = 10^5, so (10^t)^t = 10^5, giving t^2 = 5. Thus x = 10^(sqrt5) and 10^(-sqrt5). Their product is 1.
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. Let t = log_10 x so x = 10^t. Then x^(log_10 x) = 10^5, so (10^t)^t = 10^5, giving t^2 = 5. Thus x = 10^(sqrt5) and 10^(-sqrt5). Their product is 1. Trap: Since x > 0, the substitution t = log_10 x is valid and simplifies everything. 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: D