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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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