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Question Number 5734 by sanusihammed last updated on 25/May/16
Please show me an approach to solve this...    2^x  = 4x    Find the value of x
$${Please}\:{show}\:{me}\:{an}\:{approach}\:{to}\:{solve}\:{this}… \\ $$$$ \\ $$$$\mathrm{2}^{{x}} \:=\:\mathrm{4}{x} \\ $$$$ \\ $$$${Find}\:{the}\:{value}\:{of}\:{x}\: \\ $$
Answered by prakash jain last updated on 25/May/16
2^x =4x  1=((4x)/2^x )  1=((4x)/e^(xln 2) )=4xe^(−xln 2)   (1/4)=xe^(−xln 2)   ((−ln 2)/4)=−xln 2e^(−xln 2)   W(−((ln 2)/4))=−xln 2  x=−((W(−((ln 2)/4)))/(ln 2))  W is product logarithm function  W(xe^x )=x   So W(−xln 2e^(−xln 2) )=−xln 2
$$\mathrm{2}^{{x}} =\mathrm{4}{x} \\ $$$$\mathrm{1}=\frac{\mathrm{4}{x}}{\mathrm{2}^{{x}} } \\ $$$$\mathrm{1}=\frac{\mathrm{4}{x}}{{e}^{{x}\mathrm{ln}\:\mathrm{2}} }=\mathrm{4}{xe}^{−{x}\mathrm{ln}\:\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}={xe}^{−{x}\mathrm{ln}\:\mathrm{2}} \\ $$$$\frac{−\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}=−{x}\mathrm{ln}\:\mathrm{2}{e}^{−{x}\mathrm{ln}\:\mathrm{2}} \\ $$$${W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\right)=−{x}\mathrm{ln}\:\mathrm{2} \\ $$$${x}=−\frac{{W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\right)}{\mathrm{ln}\:\mathrm{2}} \\ $$$${W}\:\mathrm{is}\:\mathrm{product}\:\mathrm{logarithm}\:\mathrm{function} \\ $$$${W}\left({xe}^{{x}} \right)={x}\: \\ $$$${S}\mathrm{o}\:{W}\left(−{x}\mathrm{ln}\:\mathrm{2}{e}^{−{x}\mathrm{ln}\:\mathrm{2}} \right)=−{x}\mathrm{ln}\:\mathrm{2} \\ $$
Commented by Yozzii last updated on 26/May/16
Commented by Yozzii last updated on 26/May/16
In case you wanted a bit more  information on W.
$${In}\:{case}\:{you}\:{wanted}\:{a}\:{bit}\:{more} \\ $$$${information}\:{on}\:{W}. \\ $$
Commented by Rasheed Soomro last updated on 26/May/16
How you derived ((−ln 2)/4)=−xln 2e^(−xln 2)   from (1/4)=xe^(−xln 2)  ?  Please insert some steps more.
$${How}\:{you}\:{derived}\:\frac{−\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}=−{x}\mathrm{ln}\:\mathrm{2}{e}^{−{x}\mathrm{ln}\:\mathrm{2}} \\ $$$${from}\:\frac{\mathrm{1}}{\mathrm{4}}={xe}^{−{x}\mathrm{ln}\:\mathrm{2}} \:? \\ $$$${Please}\:{insert}\:{some}\:{steps}\:{more}. \\ $$
Commented by prakash jain last updated on 26/May/16
Multiplied both sides by −ln 2
$$\mathrm{Multiplied}\:\mathrm{both}\:\mathrm{sides}\:\mathrm{by}\:−\mathrm{ln}\:\mathrm{2} \\ $$
Commented by Rasheed Soomro last updated on 26/May/16
ThankS! I considered −xln 2e^(−xln 2)   as −xln (2e^(−xln 2) ).So couldn′t understand.
$$\mathcal{T}{hank}\mathcal{S}!\:{I}\:{considered}\:−{x}\mathrm{ln}\:\mathrm{2}{e}^{−{x}\mathrm{ln}\:\mathrm{2}} \\ $$$${as}\:−{x}\mathrm{ln}\:\left(\mathrm{2}{e}^{−{x}\mathrm{ln}\:\mathrm{2}} \right).{So}\:{couldn}'{t}\:{understand}. \\ $$

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