Question

BRAIN GAMES IS HERE

x^x=100 find x
????? If you have mind then solve for x

Answer 1

BTW it's not brain grame question. this can be solve with help of higher advanced calaculus .
______________________________
Let be try ,
we can solve this with graphically , and find the value of x € [ 3, 4 ]
see the figure .


Numerical method :-
x^x = 100
Let f(x) = x^x - 100 ----(1)
now , differentiate wrt x
f'(x) = x^x {lnx + 1}---------(2)

divide equation (1) ÷(2)
f(x)/f'(x) = {x^x - 100}/x^x(1 +lnx)
use, Newton's concept .
x(n+1) = x(n) - f{x(n)}/f'{x(n)} where, n≥0
so, x(n+1) = x(n) - (x^x -100)/x^x(1 + lnx)
Let x(0) = 4 , { 4 is smallest integer for (4)^4 > 100 }

put n = 1
x(1) = x(0) - {(4)⁴ - 100}/4⁴(1 + ln4)
= 4 - 156/256(1 + ln4)
= 3.74464

similarly,
x(2) = 3.62073
x(3) = 3.5979
x(4) =3.5972855.
x(5) = 3.597285024
x(6) = 3.597285024

hence,
approximately value of x = 3.597285024


[ it's solve with help of Newtons theory , product logarithmic etc . the chapters of calculus . you can understand when you reach in BSC or PG ]