Continuous/probability version of arithmetic/geometric mean inequality

If one has a random variable X, described by a finite probability distribution with equally likely possible values ${x}_{1},\dots ,{x}_{n}$, then the inequality: $E[\mathrm{log}X]\le \mathrm{log}(E[X])$ is a reformulation of the arithmetic/geometric mean inequality.

And a necessary and sufficient condition for equality (maybe only if the distribution is concentrated at one point, for example)?

If one has a random variable X, described by a finite probability distribution with equally likely possible values ${x}_{1},\dots ,{x}_{n}$, then the inequality: $E[\mathrm{log}X]\le \mathrm{log}(E[X])$ is a reformulation of the arithmetic/geometric mean inequality.

And a necessary and sufficient condition for equality (maybe only if the distribution is concentrated at one point, for example)?