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I was casually reading an article (in economics) which had the following approximation for $\log(E(X))$:

$\log(E(X)) \approx E(\log(X))+0.5 \mathrm{var}(\log(X))$,

which the author says is exact if X is log-normal (which I know).

What I don't know is how to derive this approximation. I tried calculating a second order Taylor approximation and all I came up with is this expression:

$\log(E(X)) \approx E(\log(X))+0.5\frac{\mathrm{var}(X)}{E(X)^2}$

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By the Delta method, the variance of a function of an RV is approximately equal to the variance of the RV times the squared derivative evaluated at the mean. Hence

$$\mathrm{var}(\log(X)) \approx \frac{1}{ \left[E(X)\right] ^2 } \mathrm{var}(X)$$

and there you have it. Your derivation was right of course.

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