Which of these statements about y = $\sqrt{8m}$ is correct?

Analyze the question: We need to take logs of y = √(8m) and see which option matches.

Concept: Log laws:

• log(ab) = log a + log b
• log(aⁿ) = n log a
• √x = x^(1/2)

Solution:

y = √(8m) = (8m)^(1/2)

Take logs of both sides:

log y = ½ log(8m)

log y = ½ (log 8 + log m)

Since 8 = 2³, log 8 = 3 log 2:

log y = ½ (3 log 2 + log m)

log y = (3/2) log 2 + ½ log m

Checking the options:

• A: Uses multiplication of logs (log 8 × log √m). This is wrong, because the product rule gives addition.
• B: Multiplies 3 log 2 by ½ log m. This is wrong for the same reason.
• C: Subtracts ½ log m. This is wrong, because the terms are added.
• D: (3/2) log 2 + ½ log m. This matches our result.

Tip: When you see a product inside a log, it becomes a sum. A square root means a power of ½, so multiply the whole log by ½. Always rewrite numbers like 8 as powers of 2 (8 = 2³) to simplify.