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Someone came to me recently with this seemingly simple equation to solve:

$$8 \log(x) - x = 0$$

So far, everything I have tried has been a dead end. Is there a symbolic solution to this kind of equation? If so, how do I get there?

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See Lambert W Function. Using WA. Regards – Amzoti Jan 25 at 22:08
This really isn't a precalculus thing anymore, although it seems to be. – DoctorBatmanGod Jan 26 at 0:04
@DoctorBatmanGod: Logarithms and inverse functions are typical pre-calculus topics. It seems plausible that solving this equation might be investigated. – robjohn♦ Jan 26 at 0:07
@robjohn I absolutely agree, but its solution involves topics not covered in precalculus. You would certainly know more than me about proper classification though, Mr. Moderator. – DoctorBatmanGod Jan 26 at 0:11
A related problem. – Mhenni Benghorbal Jan 26 at 0:25

1 Answer

up vote 1 down vote accepted

$$ 8\log(x)=x\Rightarrow -x/8\ e^{-x/8}=-1/8 $$ Therefore, $$ -x/8=\mathrm{W}(-1/8)\Rightarrow x=-8\mathrm{W}(-1/8) $$ where $\mathrm{W}$ is the Lambert W function, the inverse to $x=we^w$.

Mathematica yields $1.1553708251000778334$, and $26.093485476611910215$ as the two real solutions: N[-8 LambertW[0, -1/8], 20] and N[-8 LambertW[-1, -1/8], 20].

For those lacking a CAS like Mathematica, I have outlined an algorithm to compute Lambert W.

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Or you could just use wolfram alpha, producing this and this. – DoctorBatmanGod Jan 26 at 0:03
Wolfram Alpha is essentially a remote CAS. – robjohn♦ Jan 26 at 0:11

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