# 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?

-

3
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
– Mhenni Benghorbal Jan 26 at 0:25

## 1 Answer

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

-

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
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9097535014152527, "perplexity_flag": "middle"}