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Question-209453




Question Number 209453 by peter frank last updated on 10/Jul/24
Commented by peter frank last updated on 10/Jul/24
help B
$$\mathrm{help}\:\mathrm{B} \\ $$
Answered by mr W last updated on 10/Jul/24
((ln 2)/(ln x))=(x/(25))  xln x=25 ln 2  (ln x)e^(ln x) =25 ln 2  ⇒ln x=W(25 ln 2)  ⇒x=e^(W(25 ln 2)) =((25 ln 2)/(W(25 ln 2)))≈8.224079
$$\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:{x}}=\frac{{x}}{\mathrm{25}} \\ $$$${x}\mathrm{ln}\:{x}=\mathrm{25}\:\mathrm{ln}\:\mathrm{2} \\ $$$$\left(\mathrm{ln}\:{x}\right){e}^{\mathrm{ln}\:{x}} =\mathrm{25}\:\mathrm{ln}\:\mathrm{2} \\ $$$$\Rightarrow\mathrm{ln}\:{x}={W}\left(\mathrm{25}\:\mathrm{ln}\:\mathrm{2}\right) \\ $$$$\Rightarrow{x}={e}^{{W}\left(\mathrm{25}\:\mathrm{ln}\:\mathrm{2}\right)} =\frac{\mathrm{25}\:\mathrm{ln}\:\mathrm{2}}{{W}\left(\mathrm{25}\:\mathrm{ln}\:\mathrm{2}\right)}\approx\mathrm{8}.\mathrm{224079} \\ $$
Commented by peter frank last updated on 11/Jul/24
thank you mr W
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{mr}\:\mathrm{W} \\ $$
Answered by Ghisom last updated on 10/Jul/24
((ln 2)/(ln x))=(x/(25))  xln x =25ln 2  t=ln x  e^t t=25ln 2  t=W (25ln 2)  x=e^(W (25ln 2)) ≈8.22408
$$\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{ln}\:{x}}=\frac{{x}}{\mathrm{25}} \\ $$$${x}\mathrm{ln}\:{x}\:=\mathrm{25ln}\:\mathrm{2} \\ $$$${t}=\mathrm{ln}\:{x} \\ $$$$\mathrm{e}^{{t}} {t}=\mathrm{25ln}\:\mathrm{2} \\ $$$${t}={W}\:\left(\mathrm{25ln}\:\mathrm{2}\right) \\ $$$${x}=\mathrm{e}^{\mathrm{W}\:\left(\mathrm{25ln}\:\mathrm{2}\right)} \approx\mathrm{8}.\mathrm{22408} \\ $$
Commented by peter frank last updated on 11/Jul/24
thank you Ghison
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{Ghison} \\ $$

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