Your procedure is correct. If you want to write out things more clearly I suggest that you write down the $n$th partial sums $\begin{align} s_n &= \sum_{i=1}^{n} \log\left(\frac{i+1}{i}\right) \\ &= \sum_{i=1}^{n} \log(i+1) - \log(i) \\ &= [\log(2) - \log(1)] + \dots [\log(n+1) - \log(n)] \\ &= \log(n+1). \end{align} $ Hence $ \lim_{n \to \infty} s_n = \lim_{n\to \infty} \log(n+1) = \infty.$ So then you say that since the limit does not exist, the series is divergent by definition.
Note: The notation is important. It is not correct to write $\lim_{n\to \infty} \log(n+1) \to \infty$, we write $\lim_{n\to \infty} \log(n+1) = \infty.$