<p>The number of solutions of the equation \(\log_{x+1}(2x^2 + 7x + 5) + \log_{2x+5}((x+1)^2) - 4 = 0\), where \(x > 0\), is</p>
Step-by-Step Solution
Key Concept: Use factorization 2x^2 + 7x + 5 = (2x + 5)(x + 1). After splitting logs and using change of base, the equation reduces to one solvable branch in x > 0. Only one admissible value satisfies all domain conditions, so ...
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. Use factorization 2x^2 + 7x + 5 = (2x + 5)(x + 1). After splitting logs and using change of base, the equation reduces to one solvable branch in x > 0. Only one admissible value satisfies all domain conditions, so the number of solutions is 1. Trap: The domain x > 0 also forces both log bases to stay positive and not equal to 1. 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: A