First, I will write the expression clearly:
[tex]r= \sqrt[3]{ (A^{2}B)/C } [/tex]
[tex]r= (A^{2}B/C)^{1/3}[/tex]
Next, use these logarithm properties
log (x^n) = nlog(x)
log(xy/z) = log x + log y - log z
Then,
log r = log {(A^2)B/C}^(1/3) = (1/3) [ 2logA + logB - log C]
