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Answer 1 · 2017-03-22T10:38:16

$y = 1.4434035067785973$ $y = 1.4434035067785973$

${(4 y)}^{y} - 8^{5 y - 6} = 0$ $(4y)^y -8^(5y-6)=0$

$2^{2 y} y^{y} = 2^{3 (5 y - 6)} = 2^{15 y} 2^{- 18}$ $2^(2y)y^y=2^(3(5y-6))=2^(15y)2^(-18)$

simplifying

$y^{y} = 2^{13 y} 2^{- 18}$ $y^y=2^(13y)2^(-18)$

or

${(\frac{y}{2^{13}})}^{y} = 2^{- 18}$ $(y/2^13)^y=2^(-18)$

Calling now $z = \frac{y}{2^{13}}$ $z=y/2^(13)$

$z^{2^{13} z} = {(z^{z})}^{2^{13}} = 2^{- 18}$ $z^(2^(13)z)=(z^z)^(2^13)=2^(-18)$

or

$z^{z} = 2^{- \frac{18}{2^{13}}}$ $z^z = 2^(-18/(2^13))$

giving

$z = z_{0} = \frac{log (a)}{W (log (a))} = 0.00017619671713605924$ $z=z_0=log(a)/(W(log(a)))=0.00017619671713605924$

with $a = 2^{- \frac{18}{2^{13}}}$ $a=2^(-18/(2^13))$

where $W (z) = z e^{z}$ $W(z)=ze^z$ is the so called Lambert function.

and finally

$y = 2^{13} \times 0.00017619671713605924 = 1.4434035067785973$ $y = 2^(13) xx 0.00017619671713605924=1.4434035067785973$