The two other identities are only tricky because they're using logarithms with a different base and they're writing their roots with the radical symbol. If you

The two other identities are only tricky because they're using logarithms with a different base and they're writing their roots with the radical symbol. If you write them using log_b(x) = log(x)/log(b) they become: log(z)/log(z^(1/y)) = y z^(log(x)/log(z)) = x the first is obvious from the fact that log(x^y) = y log(x) and the latter is why log(x)/log(z) is also considered the base z logarithm. The only reason it looks nonobvious is because the notation they chose makes it non-obvious that log_b(y) = 1/log_y(b) (and that the yth root of x is x^(1/y)).