Rewrite and solve the zero-sum log equation: log_a x + log_a(1 + x) = 0. Convert to a single logarithm and find the equivalent quadratic.

Introduction / Context:
We transform a sum of logs into a single log, use the definition log_a(1) = 0, and then rewrite the condition in polynomial form.

Given Data / Assumptions:

• log_a x + log_a(1 + x) = 0
• a > 0, a ≠ 1; x > 0 and x + 1 > 0 for log arguments.

Concept / Approach:
Convert the sum to log_a( x(1 + x) ). If the sum equals 0, the argument must be 1, because log_a(1) = 0 for any valid base a.

Step-by-Step Solution:

Verification / Alternative check:
If x solves x^2 + x − 1 = 0 and satisfies domain constraints, then x(1 + x) = 1, so log_a( x(1 + x) ) = log_a(1) = 0, confirming equivalence.

Why Other Options Are Wrong:
Changing the constant term or its sign produces a different required product, not equal to 1; options with “e” are irrelevant constants here.

Common Pitfalls:
Forgetting the domain (x > 0) or failing to combine logs correctly can produce extraneous or invalid results.

Final Answer:
x2 + x - 1 = 0