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

sinbadh
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Stefan
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1 Answers1

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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.

Fly by Night
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  • Why is $W(5)$ more meaningless than, say, $\ln(5)$? – gniourf_gniourf Mar 03 '16 at 18:45
  • @gniourf_gniourf As I said: it's my personal, humble opinion. The exponential functions $x \mapsto A\mathrm e^{bx}$ make sense to me in terms of real world examples, e.g. continuously compounded interest, approximating nuclear decay or bacteria growth. The natural logarithm is the inverse of a function that I know well. Moreover, I can estimate $\ln x$ in my head to at least the same order of magnitude. I don't know of - although I'm sure there are - real world applications of the W-function, and I can't estimate it, even with a pen and paper. – Fly by Night Mar 03 '16 at 19:00