If log8 m + log8 1/6 = 2/3, then m is equal to

Concept: For logarithms with the same base, the product rule states loga(m) + loga(n) = loga(mn). Also, by the definition of a logarithm, logx(y) = a means y = xa — converting a log equation into an exponential one lets us solve for the unknown directly.

Application:

1. Combine the two logarithmic terms on the left using the product rule: log8(m) + log8(1/6) = log8(m × 1/6) = log8(m/6).

2. The equation becomes log8(m/6) = 2/3.

3. Convert to exponential form using the definition of a logarithm: m/6 = 82/3.

4. Rewrite 8 as 23 and simplify the exponent: 82/3 = (23)2/3 = 22.

5. So m/6 = 22, and solving for m gives m = 6 × 22 = 24.

Cross-check: Substitute m = 24 into the original left-hand side: log8(24) + log8(1/6) = log8(24 × 1/6) = log8(4). Writing both numbers as powers of 2 (8 = 23, 4 = 22) and using the power rule, log8(4) = 2/3 — exactly matching the right-hand side, which confirms the value of m.

∴ m = 24.