what is the solution to e^x = x?

e^x = x

1 = x / e^x

1 = x * e^(-x)

-1 =(-x) * e^(-x)

In mathematics, the Lambert W function, also called the Omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(w) = w e^w where e^w is the exponential function and w is any complex number. In other words, the defining equation for W(z) is

z = W(z) e^(W(z))

by comparing this definition to our last equation we have -x=W(z) and z=(-1)

-x=W(-1)

x= - W(-1)

x = 0.3181315 - 1.3372357 i

The Lambert W relation cannot be expressed in terms of elementary functions. It is calculated using series expansions instead.

In Matlab, function lambertw(x) evaluates principal value of Lambert's W function while lambertw(k,x) is the k-th branch of this multi-valued function.

The principal value of the Lambert W-function is implemented in Wolfram products like Mathematica and WolframAlpha as ProductLog[z]. Different branches of the function are available as ProductLog[k, z], where k is any integer and k=0 corresponds to the principal value.

Solution from the above is based on principal value of W-function (k=0). Solutions for branches other than k=0 are

k=1 ---> x=2.0622777 - 7.5886312 i

k=2 ---> x=2.6531920 - 13.9492083 i

k=3 ---> x=3.0202397 - 20.2724576 i

etc.

that doesnt really make sense because you would end up with x=ln(x) so a variable cant equal the same variable if the coefficient is different

Use a Numerical Method like Newton-Rampson .- ( either way it has No solution )

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