Members of a field named "mathematical optimization" ought to be able to solve this in a flash.
About 6.578 is the maximum. This is achieved when y = log(5)/log(2) = 2.322 and x is (somewhat surprisingly) not 1, which would solve the first equality, but when x = 0.353. There is unfortunately no closed-form solution for this second intersection of y = 5^x and y = 5*x.
Here is a Seeker model.
env = skr.Env("license.sio")
x = env.continuous(0.0001,1)
y = env.continuous(1,5)
env.enforce_eq(env.power(2,y)+env.power(8,y),130)
env.enforce_eq(env.power(5,x),5*x)
obj = y/x
env.maximize(obj, 0.1)
print("value", obj.get_value(), " x =", x.get_value(), " y =", y.get_value())
env.end()
It outputs
value 6.577906187741922 x = 0.3529889364583658 y = 2.321928109333925