If log3(x4 − x3) − log3(x − 1) = 3, then x is equal to?

Concept: Two logarithm rules apply here — the quotient rule loga(m) − loga(n) = loga(m/n), and the power rule loga(mn) = n·loga(m). Together they let a difference of same-base logs be rewritten as a single ratio, then simplified by factoring.

Application:

1. Start with the given equation: log3(x4 − x3) − log3(x − 1) = 3.

2. Apply the quotient rule to combine the logarithms: log3[(x4 − x3)/(x − 1)] = 3.

3. Factor the numerator: x4 − x3 = x3(x − 1), so the expression becomes log3[x3(x − 1)/(x − 1)].

4. Cancel the common factor (x − 1), valid since x ≠ 1: log3(x3) = 3.

5. Apply the power rule: 3·log3(x) = 3, so log3(x) = 1.

6. Convert to exponential form: x = 31 = 3.

7. Check the domain: the original equation requires x − 1 > 0 and x4 − x3 > 0, i.e. x > 1 — x = 3 satisfies this.

Cross-check: Substituting x = 3 back into the original equation: x4 − x3 = 81 − 27 = 54 and x − 1 = 2, so log3(54) − log3(2) = log3(27) = 3 — exactly matching the right-hand side.

Hence, x = 3.