log √8 / log 8 is equal to:

Concept: For any base and any real exponent a, the power rule for logarithms gives log(xa) = a · log(x). A square root is itself an exponent of one-half, i.e. √x = x1/2.

1. Rewrite the numerator's radical as a power of 8: √8 = 81/2.

2. Apply the power rule to this power: log(81/2) = 1/2 · log 8.

3. Substitute this back into the given ratio: (1/2 · log 8) / log 8.

4. Since 8 ≠ 1, log 8 ≠ 0, so the log 8 term cancels top and bottom, leaving exactly 1/2.

Cross-check: Using approximate values, log 8 ≈ 0.903 and √8 ≈ 2.828, so log √8 ≈ 0.4515. The ratio 0.4515 / 0.903 ≈ 0.5, confirming the algebraic result independently.

Result: log √8 / log 8 = 1/2, matching the option "1/2".