Is the mean of a function and function of means equal for the Log-function?

Question

I am trying to understand something that is struggling me. I have a set of data files that measures the time evolution of a physical quantities over many realizations. Where, the main meaning for a specific time $t_1$ for instance resides on the average over measured values at $t_1$. If I call $\boldsymbol{m (t_1)}$ the measure at a specific time for all the realizations. Is $\log \langle m(t_1) \rangle = \langle \log m(t_1) \rangle$ holds?

Answer 1

No. The meaning of $\langle m(t_1)\rangle$ is the mathematical expectation of the quantity $m(t_1)$ over all possible realizations. If you apply a linear function to $m(t_1)$, say, then linearity of expectation allows to pull it out. The logarithm function, however, is not linear, and so, unless $m(t_1)$ is constant, you cannot infer that your equality holds.