solve 2^3^x = 4.2^5^x
I interpret the powers from right to left, as is the usual convention. This translates the problem to:
2^(3x) = 4.2^(5x)
This is essentially an algebra problem.
take log on both sides,
(3x) log(2) = (5x) log(4.2)
Rearranging,
(3/5)x = log(4.2)/log(2)
Take log again,
x log(3/5) = log(log(4.2)/log(2))
solving for x
x = log(log(4.2)/log(2)) / log(3/5)
The numerical value should be around -1.4