## Thursday, August 05, 2021

### [xvxqfxbe] log(x)^e

below are some values of the function ln(x)^e mentioned in passing in what if?  ("it's hard to imagine a situation where it would apply").

(future post xvvojqgz: a family of sublinear functions which includes log(x)^a .)

log(15)^e ~= 15 is interesting.  exact root is at e^e=15.15 (incidentally, cannot be solved by Mathematica).  e^e is also one of the two places where the derivative (e*log(x)^(e-1))/x is equal to 1.  the other place is at e.

the function is undefined (not real) for x<1, is 0 at x=1, and is concave up until x=e^(e-1) ~= 5.6 .  beyond that it is concave down, looking like logarithm stretched vertically.

? for(x=1,100,print(x," ",log(x)^exp(1)))

1 0.E-104
2 0.36924850293727217787132470039627581740
3 1.2912987577003355870877537640846624421
4 2.4299823001842341187655754458542814676
5 3.6458500029176164395226564026808836811
6 4.8807395005561376588065054069466006352
7 6.1082556825788914851855992136543802030
8 7.3159171714026919335862289082826365792
9 8.4978899047689831684619398650361834510
10 9.6517113371265804200224873896460690389
11 10.776694259916205996785869908768082163
12 11.873102099180277287313877029048604842
13 12.941703755218028786585854204823000068
14 13.983525601880977853800827769197483163
15 14.999710436696503139405289463372545422
16 15.991436477703930673219971288091890800
17 16.959870992204533502880218064662587782
18 17.906144307266183158923973049335659051
19 18.831335982079262438706685167439483576
20 19.736468288842708966994851221338811172
21 20.622504082216135466618370299834077616
22 21.490347274782976376256109490295255178
23 22.340844818996911975597397737472055947
24 23.174789513401793884844706845017952832
25 23.992923209553188325597330639887295492
26 24.795940158192260105481827151318515476
27 25.584490335701397675386639434747133379
28 26.359182656986165551620627278325019688
29 27.120588022367859551703225350496746335
30 27.869242172368279452180382592976760343
31 28.605648340811762844794391679099373680
32 29.330279706899948055080591176599573070
33 30.043581653050506187724021976548550580
34 30.745973838773756863923115075937938965
35 31.437852102633298342330074891809315099
36 32.119590205017690618230707385587282759
37 32.791541424447505516846202861288067484
38 33.454040019724167674693737882280241000
39 34.107402569570267663189347223741498181
40 34.751929200630484726652260812153529960
41 35.387904713872129426358930589815770451
42 36.015599618591582198740260742948579039
43 36.635271082426387597322131850359803723
44 37.247163805009243651232661075291578408
45 37.851510822188441814142159501403879147
46 38.448534247083139634022877301592638589
47 39.038445953641561548250189199885856858
48 39.621448207824157278301567402942267996
49 40.197734251039035599208358464514513803
50 40.767488840010125034758084954041640851
51 41.330888746855707606144481917780176033
52 41.888103222792402108983766634150132447
53 42.439294428553621087660547215983087713
54 42.984617834318466127590373030514992718
55 43.524222591683675833518154011221976185
56 44.058251879974578508802885770530953735
57 44.586843228978271643568660443898520883
58 45.110128819990956510451475878558180123
59 45.628235766899247959317461941258362528
60 46.141286378860335477821643566750581465
61 46.649398406006281024210501077040126370
62 47.152685269471883294191926106167898592
63 47.651256276931971885547337264688160925
64 48.145216824731429503807322407868140649
65 48.634668587598527242849276937882143504
66 49.119709696848273613667680978678120758
67 49.600434907906510562017589768041002488
68 50.076935757916626773409492617364003493
69 50.549300714128276041189693988077642012
70 51.017615313710740808476641195216414902
71 51.481962295581992342627197014154358918
72 51.942421724797555410784745489549114694
73 52.399071110000527792848453034220826937
74 52.851985514395123275448810305067001632
75 53.301237660670534025507527156372222300
76 53.746898030269416046264533865944486028
77 54.189034957365595589057170412467485700
78 54.627714717888411176824901478100616112
79 55.063001613906208615635478354216624506
80 55.494958053658682467757047410454359919
81 55.923644627506815861928954476263247447
82 56.349120180049939315188607707424478897
83 56.771441878641753631099977910180028178
84 57.190665278520902393817761737703290866
85 57.606844384756710225365174176488442540
86 58.020031711196910162858965159956844053
87 58.430278336591464532607403942168049241
88 58.837633958054845644327282657582564098
89 59.242146942018301392237248924052747352
90 59.643864372813610218088835059978620352
91 60.042832099020560747266062237840011273
92 60.439094777701811016890566985708574821
93 60.832695916640833581882479112218249193
94 61.223677914691284092751583786362689154
95 61.612082100339295034328110882984171752
96 61.997948768573850254381585509194006452
97 62.381317216154500554049234305276383584
98 62.762225775360200254062034093297177176
99 63.140711846297946697078740198590997394
100 63.516811927845159317401029975041007710