No.

For example, if $X$ follows a log normal distribution, where $\log(X) \sim N(\mu,\sigma)$, then $E[\log(x)] = \mu$ and is independent of $\sigma$. However, its mean is $E[X] = \exp \left(\mu + \frac{\sigma^2}{2} \right)$. Clearly, you cannot derive a $\sigma$ dependent number form a $\sigma$ independent number.

No.

For example, if $X$ follows a log normal distribution, where $\log(X) \sim N(\mu,\sigma)$, then $E[\log(x)] = \mu$ and is independent of $\sigma$. However, its mean is $E[X] = \exp \left(\mu + \frac{\sigma^2}{2} \right)$. Clearly, you cannot derive a $\sigma$ dependent number from a $\sigma$ independent number.