Find the value of log$_{\sqrt{3}}$ 81

Analyze the question: We need the power to which √3 must be raised to get 81.

Concept: If log_b N = x, then b^x = N.

Solution:

Let x = log_{√3} 81

Then (√3)^x = 81

Write both sides as powers of 3:

• √3 = 3^(1/2), so (√3)^x = 3^(x/2)
• 81 = 3^4

So 3^(x/2) = 3^4

Equating the powers: x/2 = 4, so x = 8

Check: (√3)^8 = (3^(1/2))^8 = 3^4 = 81 ✓

Answer: C (8)

Tip: When the base is a root, convert everything to powers of the same prime number (here 3). Alternatively, use the change-of-base idea: log_{√3} 81 = log_3 81 ÷ log_3 √3 = 4 ÷ (1/2) = 8. Dividing by a half doubles the result, so a base of √3 always gives double the log to base 3.