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

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 '13 at 22:08
This really isn't a precalculus thing anymore, although it seems to be. – Sam DeHority Jan 26 '13 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 '13 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. – Sam DeHority Jan 26 '13 at 0:11
– Mhenni Benghorbal Jan 26 '13 at 0:25

$$8\log(x)=x\Rightarrow -x/8\ e^{-x/8}=-1/8$$ 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].