# Solve ((4y)/8^5)^y = 8^-6 ?

Apr 16, 2017

$y = 1.4434$

#### Explanation:

Manipulating the original equation after doing $z = \frac{4 y}{8} ^ 5$ we arrive at

${z}^{\left({8}^{5} / 4\right) z} = {8}^{-} 6$ or

${\left({z}^{\left({8}^{5} / 4\right) z}\right)}^{\frac{4}{8} ^ 5} = {\left({8}^{-} 6\right)}^{\frac{4}{8} ^ 5}$ or

${z}^{z} = {8}^{- \frac{24}{8} ^ 5}$

Now we have the format to use the Lambert $W$ function to solve the equation.

https://en.wikipedia.org/wiki/Lambert_W_function

so with the help of https://www.wolframalpha.com/

$z = - \frac{9 {\log}_{e} \left(2\right)}{{2}^{12} W \left(- \frac{9 {\log}_{e} \left(2\right)}{2} ^ 12\right)} = 0.000176197$

after that we have

$y = {2}^{13} z = 1.4434$