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Empower With X and 0.4 (Posted on 2008-03-10) Difficulty: 2 of 5
Determine all possible positive real numbers X that satisfy this equation:

X0.4X = 0.4X0.4

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

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution computer exploration (spoiler) | Comment 1 of 7

It seems, from the following tables, that 0.4 is the only positive real that will work:

    x     left side right side
0.0400000 0.0449127 0.7765869
0.0800000 0.0956358 0.7163165
0.1200000 0.1496449 0.6754481
0.1600000 0.2054240 0.6438872
0.2000000 0.2618607 0.6179594
0.2400000 0.3180995 0.5958614
0.2800000 0.3734754 0.5765597
0.3200000 0.4274739 0.5594007
0.3600000 0.4797027 0.5439425
0.4000000 0.5298707 0.5298707
0.4400000 0.5777702 0.5169531
0.4800000 0.6232621 0.5050128
0.5200000 0.6662640 0.4939116
0.5600000 0.7067397 0.4835397
0.6000000 0.7446900 0.4738076
0.6400000 0.7801455 0.4646421
0.6800000 0.8131599 0.4559819
0.7200000 0.8438051 0.4477753
0.7600000 0.8721661 0.4399785
0.8000000 0.8983376 0.4325537
0.8400000 0.9224211 0.4254683
0.8800000 0.9445218 0.4186938
0.9200000 0.9647470 0.4122052
0.9600000 0.9832043 0.4059804
1.0000000 1.0000000 0.4000000

1.0666667 1.0245829 0.3905290
1.1333333 1.0453038 0.3816189
1.2000000 1.0625981 0.3732109
1.2666667 1.0768688 0.3652550
1.3333333 1.0884845 0.3577084
1.4000000 1.0977796 0.3505341
1.4666667 1.1050551 0.3436998
1.5333333 1.1105805 0.3371774
1.6000000 1.1145959 0.3309421
1.6666667 1.1173139 0.3249718
1.7333333 1.1189228 0.3192472
1.8000000 1.1195880 0.3137505
1.8666667 1.1194550 0.3084663
1.9333333 1.1186510 0.3033803
2.0000000 1.1172871 0.2984798
2.0666667 1.1154603 0.2937532
2.1333333 1.1132546 0.2891900
2.2000000 1.1107431 0.2847804
2.2666667 1.1079888 0.2805158
2.3333333 1.1050462 0.2763880
2.4000000 1.1019620 0.2723895
2.4666667 1.0987766 0.2685136
2.5333333 1.0955241 0.2647538
2.6000000 1.0922339 0.2611044
2.6666667 1.0889307 0.2575599
2.7333333 1.0856353 0.2541152
2.8000000 1.0823654 0.2507656
2.8666667 1.0791356 0.2475067
2.9333333 1.0759578 0.2443345
3.0000000 1.0728420 0.2412451
3.0666667 1.0697960 0.2382350
3.1333333 1.0668262 0.2353006
3.2000000 1.0639375 0.2324390
3.2666667 1.0611334 0.2296470
3.3333333 1.0584167 0.2269220
3.4000000 1.0557889 0.2242612
3.4666667 1.0532510 0.2216622
3.5333333 1.0508034 0.2191227
3.6000000 1.0484458 0.2166403
3.6666667 1.0461775 0.2142130
3.7333333 1.0439973 0.2118388
3.8000000 1.0419038 0.2095158
3.8666667 1.0398955 0.2072423
3.9333333 1.0379704 0.2050164

 4.0000000 1.0361264 0.2028365
 5.0000000 1.0166172 0.1747672
 6.0000000 1.0073660 0.1531614
 7.0000000 1.0031933 0.1359327
 8.0000000 1.0013637 0.1218355
 9.0000000 1.0005762 0.1100705
10.0000000 1.0002415 0.1000967
11.0000000 1.0001006 0.0915327
12.0000000 1.0000417 0.0841004
13.0000000 1.0000172 0.0775918
14.0000000 1.0000071 0.0718479
15.0000000 1.0000029 0.0667444
16.0000000 1.0000012 0.0621830
17.0000000 1.0000005 0.0580845
18.0000000 1.0000002 0.0543848
19.0000000 1.0000001 0.0510307
20.0000000 1.0000000 0.0479784
21.0000000 1.0000000 0.0451911
22.0000000 1.0000000 0.0426375
23.0000000 1.0000000 0.0402912
24.0000000 1.0000000 0.0381296
25.0000000 1.0000000 0.0361332
26.0000000 1.0000000 0.0342849
27.0000000 1.0000000 0.0325703
28.0000000 1.0000000 0.0309763
29.0000000 1.0000000 0.0294917
30.0000000 1.0000000 0.0281065
31.0000000 1.0000000 0.0268118
32.0000000 1.0000000 0.0256000
33.0000000 1.0000000 0.0244639
34.0000000 1.0000000 0.0233974
35.0000000 1.0000000 0.0223949
36.0000000 1.0000000 0.0214512
37.0000000 1.0000000 0.0205620
38.0000000 1.0000000 0.0197230
39.0000000 1.0000000 0.0189307
40.0000000 1.0000000 0.0181815

The LHS is monotonically increasing from just above x=zero to about x=1.8, when the LHS is 1.1195880 and meanwhile the RHS has decreased from 0.7765869 to 0.3137505 for those same x values (of course at zero, the RHS is 1). They cross, of course, at x=0.4 as both are then 0.4^0.4^0.4.

From that maximum, the LHS approaches 1 asymptotically, while the RHS approaches zero, so the two curves would seem never to cross again.

Edited on March 10, 2008, 4:18 pm
  Posted by Charlie on 2008-03-10 16:15:52

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