Given log₁₀2 = 0.3010 and log₁₀3 = 0.4771, evaluate log₅512.

Concept / Approach

• Use change of base: log₅512 = log₁₀512 / log₁₀5.

Step-by-step calculation
log₁₀512 = log₁₀(2⁹) = 9 × 0.3010 = 2.709log₁₀5 = 1 − log₁₀2 = 1 − 0.3010 = 0.6990log₅512 = 2.709 / 0.6990 = 3.876 (approx.)

Verification
5³ = 125 < 512 < 625 = 5⁴ ⇒ answer between 3 and 4, consistent.