User blog comment:Sbiis Saibian/Googology101 - Part II/@comment-10429372-20141024190431/@comment-2033667-20141024212633

Here's how we can find the solutions to \(x = 10^x\) ourselves, without using WolframAlpha. The key element here is the "Lambert W function" \(W(z)\), which is defined as the solutions to the equation \(z = We^W\). As such it's actually a multifunction with many "branches." To use the Lambert W function, we coerce \(x = 10^x\) into the format \(z = We^W\) like so:

\[x = 10^x\]

\[1 = xe^{-x \ln 10}\]

\[-\ln 10 = -(x \ln 10) e^{-x \ln 10}\]

This is now in "Lambert form" (a term which I just made up), so we use the fact that \(z = ye^y \Rightarrow W(z) = y\):

\[-x \ln 10 = W(-\ln 10)\]

\[x = -\frac{W(-\ln 10)}{\ln 10}\]

Over the real numbers, \(W(z)\) is only defined for \(z < -1/e\), so clearly there are no real solutions to the original equation. Over complex numbers, \(W(z)\) has a countably infinite number of solutions for all nonzero \(z\), each falling within a branch indexed by an integer.

W cannot be expressed using elementary functions, but it's in many mathematical libraries. (It's usually computed with Newton's method or the like.) Here's how to compute it in Sage, using the "principal branch" (indexed by zero) of the W function:

sage: (-lambert_w(0, -ln(10))/ln(10)).n -0.119193073414548 - 0.750583293932439*I sage: 10^_ -0.119193073414548 - 0.750583293932440*I