Explanation:
we need to use a logarithm.
the easiest in this case is the logarithm to the base of 278.
when we apply this to both sides, we get
log278(278^4x) = log278(37800)
4x = log278(37800)
we know
loga(b) = logc(b)/logc(a)
on our calculators we usually have the logarithm to the base of 10 (log) or to the base of e (ln).
so, you can pick c = 10 or c = e. it works the same in both cases.
I pick c = 10.
so,
4x = log(37800)/log(278)
x = log(37800)/log(278)/4 = log(37800)/(4×log(278)) =
= 0.468229122... ≈ 0.4682