Question

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

Answer 1

`y = 1.4434`

Explanation:

Manipulating the original equation after doing `z = (4y)/8^5` we arrive at

`z^((8^5/4)z)=8^-6` or

`(z^((8^5/4)z))^(4/8^5)=(8^-6)^(4/8^5)` or

`z^z = 8^(-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=-(9log_e(2))/(2^12 W(-(9log_e(2))/2^12))=0.000176197`

after that we have

`y=2^13 z = 1.4434`