Question

Solve the following? `x^(1/x)=1.2`

Answer 1

Use Lambert `W` function...

Explanation:

Given:

`x^(1/x) = 1.2 = 6/5`

Raise both sides to the power `x` to get:

`x = (6/5)^x = e^(x ln(6/5))`

Multiply both sides by `e^(-x ln(6/5))` to get:

`x e^(-x ln(6/5)) = 1`

Multiply both sides by `-ln(6/5)` to get:

`-x ln(6/5) e^(-x ln(6/5)) = -ln(6/5)`

This is in the form:

`z e^z = c" "` with `z = -x ln(6/5)`

This has solutions given by the Lambert `W` function:

`-x ln(6/5) = W(-ln(6/5))`

So:

`x = -1/ln(6/5) W(-ln(6/5)) = 1/ln(5/6) W(ln(5/6))`

Actually the Lambert `W` function has multiple branches. The real solutions in this example are for `W_0` and `W_(-1)` giving:

`x = 1/ln(5/6) W_0(ln(5/6)) ~~ 1.2577345`

`x = 1/ln(5/6) W_(-1)(ln(5/6)) ~~ 14.767458`