A quick math problem came by my cube today that was worth sharing. I was asked if there was a closed-form solution to Equation 1.

Eq. 1 |

This problem does not have a closed form solution using the "everyday" functions, but it is solvable using Lambert's W function. Recall that Lambert's W function has the property shown in Equation 2.

Eq. 2 |

We can hammer equation 1 into the form of Equation 2 by the process shown in Equation 3.

Eq. 3 | |

, definition of W function | |

We now have a closed form solution, but it requires the use of a relatively uncommon function. Let's compute the numeric solution. As shown in Figure 1, I drop into Mathcad for this part.