Convex function of an Expectation of a Random Variable Intuition

Question

In working with estimating the parameters for a log-normal distribution, I initially made the mistake in parameterising the distribution using the log of the mean observation value, as opposed to taking the mean of the log-observation values. i.e. I realised: $$ E[\log(X)] \ne \log(E[X]) $$ I understand the above inequality mathematically, and in its relation to Jenson's inequality for convex functions. But is there an intuitive meaning to the RHS term, in distinction to the LHS or otherwise, or is it mathematically/statistically uninformative?