Determine all possible positive real numbers X that satisfy this equation:

X^0.4X = 0.4^X0.4

Note: For the purposes of the problem, a^bc is equal to a^(b^c)

(In reply to computer exploration (spoiler) by Charlie)

DEFDBL A-Z
df = .04
FOR x = df TO 1 STEP df
 a = x ^ (.4 ^ x): b = .4 ^ (x ^ .4)
 PRINT USING "##.#######"; x; a; b
NEXT
DO: LOOP UNTIL INKEY$ > ""
df = 3 / 45
FOR x = df TO 4 STEP df
 a = x ^ (.4 ^ x): b = .4 ^ (x ^ .4)
 PRINT USING "##.#######"; x; a; b
NEXT
DO: LOOP UNTIL INKEY$ > ""
df = 1
FOR x = df TO 40 STEP df
 a = x ^ (.4 ^ x): b = .4 ^ (x ^ .4)
 PRINT USING "##.#######"; x; a; b
NEXT

Posted by Charlie on 2008-03-10 16:21:10