Solve for x in the base-4 logarithmic equation: 2·log₄(x) = 1 + log₄(x − 1). State domain conditions and the final solution.

Introduction / Context:
We solve a logarithmic equation involving base 4 by converting the left-hand side with the power rule and equating arguments after combining using log identities.

Given Data / Assumptions:

• Equation: 2·log₄(x) = 1 + log₄(x − 1)
• Domain: x > 0 and x − 1 > 0 ⇒ x > 1

Concept / Approach:
Use 2·log₄ x = log₄(x^2) and 1 = log₄ 4. Then sum on the right: log₄ 4 + log₄(x − 1) = log₄(4(x − 1)). Equate arguments of equal logs.

Step-by-Step Solution:

Verification / Alternative check:
Check domain: x = 2 > 1; substitute: 2·log₄ 2 = 2·(1/2) = 1; right side 1 + log₄ 1 = 1 + 0 = 1. Holds exactly.

Why Other Options Are Wrong:
1 violates the domain; 3 and 4 do not satisfy the quadratic; “no real solution” is incorrect since x = 2 is valid.

Common Pitfalls:
Forgetting to use log₄ 4 = 1 or mishandling the power rule can lead to an incorrect linear equation instead of the correct quadratic.

Final Answer:
2