A Little Drive-By Math Incident

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 \ln ({{x}^{x}})=4

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 z~=~W\left( z \right)\cdot {{e}^{W(z)}}

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

Eq. 3 x\cdot \ln (x)=4
{{e}^{\ln (x)}}\cdot \ln (x)=4
{{e}^{W(4)}}\cdot W(4)=4, definition of W function
\text{Let }W(4)=\ln (x)
\therefore x={{e}^{W(4)}}

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.

Figure 1: Numerical Solution in Mathcad.

Figure 1: Numerical Solution in Mathcad.

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About mathscinotes

I am an engineer who encounters interesting math and science problems almost every day. I am not talking about BIG math here. These are everyday problems where a little bit of math really goes a long way. I thought I would write some of them down and see if others also found them interesting.
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