Determine the value of log3√2(1/18) = ?

Concept: By the definition of a logarithm, for logb(x), if the argument x can be written as the base b raised to some power n (that is, x = bn), then logb(x) equals that exponent n directly — a logarithm simply measures the power to which the base must be raised to produce the argument.

Application: Here the base is 3√2 and the argument is 1/18, so the goal is to express 1/18 as (3√2) raised to some power:

1. Square the base: (3√2)2 = 32 × (√2)2 = 9 × 2 = 18.

2. Since 18 = (3√2)2, its reciprocal 1/18 equals (3√2)-2 (a negative exponent inverts the base's power).

3. Substitute into the original expression: log3√2(1/18) = log3√2((3√2)-2).

4. Apply the logarithm definition with n = -2: log3√2((3√2)-2) = -2.

Cross-check: (3√2)2 = 9 × 2 = 18 confirms the base-power relationship used above, and raising the base 3√2 to the power -2 gives back 1/18, verifying the result.

Therefore, log3√2(1/18) = -2.