Question

How do you use the law of logarithms to calculate for u^c=(a^r)^c?

Answer 1

not sure what you want to do, but taking logs,
log(u^c) = c logu
(a^r)^c = a^(rc)
so log((a^r)^c) = rc loga

c logu = rc loga
logu = r loga
r = log_{_a}u

but even without logs, if a^c = b^c, then a=b
so, u = a^r