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Question-76176




Question Number 76176 by Master last updated on 24/Dec/19
Commented by mr W last updated on 24/Dec/19
(3x+1)8^x =6  (3x+1)2^(3x) =6  (3x+1)2^(3x+1) =12  (3x+1)e^((3x+1)ln 2) =12  (3x+1)ln 2 e^((3x+1)ln 2) =12 ln 2  (3x+1)ln 2 =W(12 ln 2)  ⇒x=(1/3)[((W(12 ln 2))/(ln 2))−1]  ≈(1/3)[((1.629884547)/(ln 2))−1]=0.450475
$$\left(\mathrm{3}{x}+\mathrm{1}\right)\mathrm{8}^{{x}} =\mathrm{6} \\ $$$$\left(\mathrm{3}{x}+\mathrm{1}\right)\mathrm{2}^{\mathrm{3}{x}} =\mathrm{6} \\ $$$$\left(\mathrm{3}{x}+\mathrm{1}\right)\mathrm{2}^{\mathrm{3}{x}+\mathrm{1}} =\mathrm{12} \\ $$$$\left(\mathrm{3}{x}+\mathrm{1}\right){e}^{\left(\mathrm{3}{x}+\mathrm{1}\right)\mathrm{ln}\:\mathrm{2}} =\mathrm{12} \\ $$$$\left(\mathrm{3}{x}+\mathrm{1}\right)\mathrm{ln}\:\mathrm{2}\:{e}^{\left(\mathrm{3}{x}+\mathrm{1}\right)\mathrm{ln}\:\mathrm{2}} =\mathrm{12}\:\mathrm{ln}\:\mathrm{2} \\ $$$$\left(\mathrm{3}{x}+\mathrm{1}\right)\mathrm{ln}\:\mathrm{2}\:={W}\left(\mathrm{12}\:\mathrm{ln}\:\mathrm{2}\right) \\ $$$$\Rightarrow{x}=\frac{\mathrm{1}}{\mathrm{3}}\left[\frac{{W}\left(\mathrm{12}\:\mathrm{ln}\:\mathrm{2}\right)}{\mathrm{ln}\:\mathrm{2}}−\mathrm{1}\right] \\ $$$$\approx\frac{\mathrm{1}}{\mathrm{3}}\left[\frac{\mathrm{1}.\mathrm{629884547}}{\mathrm{ln}\:\mathrm{2}}−\mathrm{1}\right]=\mathrm{0}.\mathrm{450475} \\ $$
Commented by Master last updated on 24/Dec/19
thanks
$$\mathrm{thanks} \\ $$
Commented by benjo last updated on 25/Dec/19
sir how to find value of W(12ln12)?   please write the formula
$$\mathrm{sir}\:\mathrm{how}\:\mathrm{to}\:\mathrm{find}\:\mathrm{value}\:\mathrm{of}\:\mathrm{W}\left(\mathrm{12ln12}\right)?\: \\ $$$$\mathrm{please}\:\mathrm{write}\:\mathrm{the}\:\mathrm{formula} \\ $$
Commented by mr W last updated on 25/Dec/19
W function′s value can only be  approximated, see wikipedia for  more about numerical evaluation.
$${W}\:{function}'{s}\:{value}\:{can}\:{only}\:{be} \\ $$$${approximated},\:{see}\:{wikipedia}\:{for} \\ $$$${more}\:{about}\:{numerical}\:{evaluation}. \\ $$

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