Question

We have `xoy=x^(xlog_e y),forall x,yin[1,oo)`.Find `x` for `x o x o x=125`?

Answer 1

`x = e^root(4)(3 log 5)`

Explanation:

Considering that for `x > 0 rArr x = e^(log x)`

and defining `x@y = e^(logx logy)`

we have

`x@x@x = e^(Log(e^(Log(e^(Log^2x)) Logx)) Logx) = ((e^(Log^2x))^Logx)^Logx`

then

`((e^(Log^2x))^Logx)^Logx=5^3`

now applying `log` to both sides

`logx log(e^(Log^2x))^Logx=log^2x log(e^(Log^2x))=log^4x = 3 log 5`

then

`log x= root(4)(3 log 5)` and

`x = e^root(4)(3 log 5)`