If x > 1 and (log₁₀x)² = 3 log₁₀x + 4, then x equals

Let t = log₁₀x. The equation becomes t² = 3t + 4, i.e. t² − 3t − 4 = 0 → (t − 4)(t + 1) = 0 → t = 4 or t = −1.

t = −1 gives x = 10⁻¹ = 0.1, which is less than 1 and rejected. So t = 4, giving x = 10⁴ = 10,000.

Tip: Substituting t = log x turns a log-quadratic into an ordinary quadratic; always check which root satisfies any stated range for x.