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8x-16-x-x-




Question Number 121850 by Khalmohmmad last updated on 12/Nov/20
8x=16^x   x=?
$$\mathrm{8}{x}=\mathrm{16}^{{x}} \\ $$$${x}=? \\ $$
Answered by mr W last updated on 12/Nov/20
8x=e^(xln 16)   8x×e^(−xln 16) =1  (−xln 16)×e^(−xln 16) =−((ln 16)/8)=−((ln 2)/2)  −x ln 16=W(−((ln 2)/2))  ⇒x=−(1/(4ln 2))W(−((ln 2)/2))= { ((1/2)),((1/4)) :}
$$\mathrm{8}{x}={e}^{{x}\mathrm{ln}\:\mathrm{16}} \\ $$$$\mathrm{8}{x}×{e}^{−{x}\mathrm{ln}\:\mathrm{16}} =\mathrm{1} \\ $$$$\left(−{x}\mathrm{ln}\:\mathrm{16}\right)×{e}^{−{x}\mathrm{ln}\:\mathrm{16}} =−\frac{\mathrm{ln}\:\mathrm{16}}{\mathrm{8}}=−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}} \\ $$$$−{x}\:\mathrm{ln}\:\mathrm{16}={W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{1}}{\mathrm{4ln}\:\mathrm{2}}{W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}}\right)=\begin{cases}{\frac{\mathrm{1}}{\mathrm{2}}}\\{\frac{\mathrm{1}}{\mathrm{4}}}\end{cases} \\ $$
Commented by BeeMee last updated on 12/Nov/20
dont understand the third line please
$${dont}\:{understand}\:{the}\:{third}\:{line}\:{please} \\ $$
Commented by mr W last updated on 12/Nov/20
8x×e^(−xln 16) =1  x×e^(−xln 16) =(1/8)  (−x ln 16)×e^(−xln 16) =−((ln 16)/8)  (−x ln 16)×e^(−xln 16) =−((ln 2)/2)  −x ln 16=W(−((ln 2)/2))  ...
$$\mathrm{8}{x}×{e}^{−{x}\mathrm{ln}\:\mathrm{16}} =\mathrm{1} \\ $$$${x}×{e}^{−{x}\mathrm{ln}\:\mathrm{16}} =\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\left(−{x}\:\mathrm{ln}\:\mathrm{16}\right)×{e}^{−{x}\mathrm{ln}\:\mathrm{16}} =−\frac{\mathrm{ln}\:\mathrm{16}}{\mathrm{8}} \\ $$$$\left(−{x}\:\mathrm{ln}\:\mathrm{16}\right)×{e}^{−{x}\mathrm{ln}\:\mathrm{16}} =−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}} \\ $$$$−{x}\:\mathrm{ln}\:\mathrm{16}={W}\left(−\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}}\right) \\ $$$$… \\ $$

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