If log_x 4 = 0.4, find the value of x.

Introduction / Context:
The expression uses a variable base x. The definition log_x 4 = 0.4 means x^0.4 = 4. We solve for x by raising to a reciprocal power or by converting to an exponential equation in base 10 or e.

Given Data / Assumptions:

• log_x 4 = 0.4.
• x > 0, x ≠ 1 (valid logarithm base).

Concept / Approach:

• By definition, log_x 4 = y ⇔ x^y = 4.
• Thus x^0.4 = 4 ⇒ x = 4^(1/0.4) = 4^(2.5).

Step-by-Step Solution:

Verification / Alternative check:
Check: log_32 4 = ln 4 / ln 32 = (2 ln 2) / (5 ln 2) = 2/5 = 0.4. Correct.

Why Other Options Are Wrong:

• 4, 16, 1 do not satisfy x^0.4 = 4.

Common Pitfalls:

• Interpreting log x4 as log( x^4 ) in base 10; the intended meaning is base x.

Final Answer:
32