If log5(x^2 + x) − log5 x = 2, find x.

Introduction / Context:Use the quotient law of logarithms to combine the difference of logs, then exponentiate to solve. Ensure arguments are positive and domain constraints are respected (x > 0, x^2 + x > 0).

Given Data / Assumptions:

• log5(x^2 + x) − log5 x = 2.
• x > 0 and x^2 + x > 0 (automatic for x > 0).

Concept / Approach:

• log5(A) − log5(B) = log5(A/B).
• If log5 T = 2, then T = 5^2 = 25.

Step-by-Step Solution:

Verification / Alternative check:Plug in x = 24: LHS = log5(24^2 + 24) − log5(24) = log5(600) − log5(24) = log5(25) = 2. Correct.

Why Other Options Are Wrong:

• 25, 23, 120 do not satisfy x + 1 = 25.

Common Pitfalls:

• Forgetting to simplify (x^2 + x)/x to x + 1.

Final Answer:24