How can I solve this equation: $xe^{ax}=b$?

Question

How can I solve this type of equation? $xe^{ax}=b$?

Answer 1

For most values of $a$ and $b$ there is no solution in terms of "elementary functions", i.e. you can rearrange to get $x$ on its own using just algebra and the functions that you learned about in school.

One way would be to use numerical methods, e.g. interval bisection, the Newton-Raphson method, linear interpolation, etc.

Another way would be to use a function called Lambert's W-function. This is a very exotic function. It is defined to be the inverse of $z \mapsto z\mathrm e^z$. You will probably need a computer programme - I've never found a calculator with this function - to make any calculations.

There are lots of examples of the W-funtion's application on Wikipedia.

Personally, in my humble opinion, using the W-function feels like cheating. I'd use a numerical method. After all, and answer in terms of the W-function, say $\mathrm W(5)$ is meaningless to a human being and we'd need to ask a computer to give a decimal approximation anyway.