A Lambert W-function with no real solution?

Question

Recently I have created this question:

$$(x+1)^xe^{x+1}=(2x)^x$$

And the solution

$$(x+1)^xe^{x+1}=(2x)^x$$$$(x+1)e^{1+\frac 1 x}=2x$$$$(1+\frac 1 x)e^{1+\frac 1 x}=2$$$$1+\frac 1 x=W(2)$$ $$x=\frac 1 {W(2)-1}$$

But Desmos shows no solution, can anyone explain what is wrong here?

$W(2)<1$ and powers of negatives are undefined in the reals. — , Jan 02 '21 at 14:43
You need to define the powers of negatives and complex then. — , Jan 03 '21 at 10:59

score 1 · Answer 1 · answered Jan 02 '21 at 14:47

1

Wolfy says the solution to $ze^z=2$ is about $z≈0.85260550201372549135$ so $x≈-6.7845137617899467654$.

The problem is defining the specified powers.

answered Jan 02 '21 at 14:47

marty cohen

1 Answers1