If log 2 = 0.3010 and log 3 = 0.4771, the value of log5 512 is:

Concept: To evaluate a logarithm in a base other than 10, use the change-of-base formula logb(a) = log(a) / log(b), computed with common (base-10) logarithms. It also helps to note that log(10/n) = log 10 − log n, so with log 10 = 1, log 5 = 1 − log 2.

1. Write 512 as a power of 2: 512 = 29.

2. Apply the change-of-base formula: log5(512) = log(512) / log(5).

3. Compute the numerator: log(512) = log(29) = 9 × log 2 = 9 × 0.3010 = 2.709.

4. Compute the denominator: log 5 = log(10/2) = log 10 − log 2 = 1 − 0.3010 = 0.699.

5. Divide: log5(512) = 2.709 / 0.699 ≈ 3.876.

Cross-check: raising 5 to the power 3.876 should reproduce 512 — 53.876 = 10(3.876 × log 5) = 10(3.876 × 0.699) = 102.708 ≈ 510.7, which matches 512 within the rounding of the given four-decimal logarithm values, confirming the result.