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Question Number 77573 by MJS last updated on 08/Jan/20

$$x\ln x=\frac{143851}{40000}$$$$\mathrm{solve}\mathrm{for}x\Rightarrow \mathrm{nice}\mathrm{surprise}$$

Commented by Tony Lin last updated on 08/Jan/20

$$lnx=\frac{143851}{40000}{x}^{-1}$$$$e^{\frac{143851}{40000}{x}^{-1}}=x$$$$x^{-1}{e}^{\frac{143851}{40000}{x}^{-1}}=1$$$$\frac{143851}{40000}{x}^{-1}{e}^{\frac{143851}{40000}{x}^{-1}}=\frac{143851}{40000}$$$$\frac{143851}{40000}{x}^{-1}=\mathbb{W}\left(\frac{143851}{40000}\right)$$$$x=\frac{\frac{143851}{40000}}{\mathbb{W}\left(\frac{143851}{40000}\right)}\fallingdotseq \frac{3.596275}{1.14473}=3.141592$$$$\fallingdotseq \pi$$

Commented by MJS last updated on 08/Jan/20

$$\mathrm{yes}.\mathrm{somebody}\mathrm{tried}\mathrm{to}\mathrm{prove}\mathrm{to}\mathrm{me}\mathrm{that}\pi$$$$\mathrm{was}\mathrm{somehow}\mathrm{rational}\mathrm{with}\mathrm{these}‘‘{\mathrm{equations}}^{″}$$$$p=\pi \ln \pi =\frac{143851}{40000}$$$$q=\pi \ln \left(\pi \ln \pi \right)=\frac{100523}{25000}$$$$\mathrm{but}\mathrm{he}\mathrm{used}\mathrm{a}\mathrm{calculator}\mathrm{with}\mathrm{only}10\mathrm{digits}$$$$p=3.5962749997...\approx 3.596275$$$$q=4.0209199997...\approx 4.02092$$$$\mathrm{near}\mathrm{enough}\mathrm{is}\mathrm{not}\mathrm{good}\mathrm{enough},\mathrm{snyway}\mathrm{I}$$$$\mathrm{I}\mathrm{found}\mathrm{it}\mathrm{interesting}$$

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