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 Empower With X and 0.4 (Posted on 2008-03-10)
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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 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 side0.0400000 0.0449127 0.77658690.0800000 0.0956358 0.71631650.1200000 0.1496449 0.67544810.1600000 0.2054240 0.64388720.2000000 0.2618607 0.61795940.2400000 0.3180995 0.59586140.2800000 0.3734754 0.57655970.3200000 0.4274739 0.55940070.3600000 0.4797027 0.54394250.4000000 0.5298707 0.52987070.4400000 0.5777702 0.51695310.4800000 0.6232621 0.50501280.5200000 0.6662640 0.49391160.5600000 0.7067397 0.48353970.6000000 0.7446900 0.47380760.6400000 0.7801455 0.46464210.6800000 0.8131599 0.45598190.7200000 0.8438051 0.44777530.7600000 0.8721661 0.43997850.8000000 0.8983376 0.43255370.8400000 0.9224211 0.42546830.8800000 0.9445218 0.41869380.9200000 0.9647470 0.41220520.9600000 0.9832043 0.40598041.0000000 1.0000000 0.4000000`
`1.0666667 1.0245829 0.39052901.1333333 1.0453038 0.38161891.2000000 1.0625981 0.37321091.2666667 1.0768688 0.36525501.3333333 1.0884845 0.35770841.4000000 1.0977796 0.35053411.4666667 1.1050551 0.34369981.5333333 1.1105805 0.33717741.6000000 1.1145959 0.33094211.6666667 1.1173139 0.32497181.7333333 1.1189228 0.31924721.8000000 1.1195880 0.31375051.8666667 1.1194550 0.30846631.9333333 1.1186510 0.30338032.0000000 1.1172871 0.29847982.0666667 1.1154603 0.29375322.1333333 1.1132546 0.28919002.2000000 1.1107431 0.28478042.2666667 1.1079888 0.28051582.3333333 1.1050462 0.27638802.4000000 1.1019620 0.27238952.4666667 1.0987766 0.26851362.5333333 1.0955241 0.26475382.6000000 1.0922339 0.26110442.6666667 1.0889307 0.25755992.7333333 1.0856353 0.25411522.8000000 1.0823654 0.25076562.8666667 1.0791356 0.24750672.9333333 1.0759578 0.24433453.0000000 1.0728420 0.24124513.0666667 1.0697960 0.23823503.1333333 1.0668262 0.23530063.2000000 1.0639375 0.23243903.2666667 1.0611334 0.22964703.3333333 1.0584167 0.22692203.4000000 1.0557889 0.22426123.4666667 1.0532510 0.22166223.5333333 1.0508034 0.21912273.6000000 1.0484458 0.21664033.6666667 1.0461775 0.21421303.7333333 1.0439973 0.21183883.8000000 1.0419038 0.20951583.8666667 1.0398955 0.20724233.9333333 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 0.110070510.0000000 1.0002415 0.100096711.0000000 1.0001006 0.091532712.0000000 1.0000417 0.084100413.0000000 1.0000172 0.077591814.0000000 1.0000071 0.071847915.0000000 1.0000029 0.066744416.0000000 1.0000012 0.062183017.0000000 1.0000005 0.058084518.0000000 1.0000002 0.054384819.0000000 1.0000001 0.051030720.0000000 1.0000000 0.047978421.0000000 1.0000000 0.045191122.0000000 1.0000000 0.042637523.0000000 1.0000000 0.040291224.0000000 1.0000000 0.038129625.0000000 1.0000000 0.036133226.0000000 1.0000000 0.034284927.0000000 1.0000000 0.032570328.0000000 1.0000000 0.030976329.0000000 1.0000000 0.029491730.0000000 1.0000000 0.028106531.0000000 1.0000000 0.026811832.0000000 1.0000000 0.025600033.0000000 1.0000000 0.024463934.0000000 1.0000000 0.023397435.0000000 1.0000000 0.022394936.0000000 1.0000000 0.021451237.0000000 1.0000000 0.020562038.0000000 1.0000000 0.019723039.0000000 1.0000000 0.018930740.0000000 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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