Suppose $X$ is distributed $N(\mu, \sigma^2)$ where $\mu \neq 0$.
Can I use the Delta Method to say that $log(X)$ ~ $N(log(\mu), \sigma^2/\mu^2)$?
It is not the case.
For $\log(X)$ to be normal, $X$ must be lognormal.
(Consider: if $Z=\log(X)$ is normal, then $X=\exp(Z)$ ... and when you exponentiate a normal random variable, what you get is called a lognormal random variable.)
More generally, taking logs "pulls in" more extreme values on the right (high values) relative to the median, while values at the far left (low values) tend to get stretched back. So if it's symmetric before taking logs, it will be relatively left skew after. This is a simple consequence of the shape of the function $\log(x)$:
(the line is tangent to the curve. In general it doesn't necessarily go close to the origin, that's just an artifact of the particular value of $m$ in this case)
Values very close to the median (indicated by an $m$ in the plot) will experience an approximately linear rescaling (the dashed blue line). Values far above $m$ will be pulled back toward $m$ relative to that rescaling experienced by the middle values, while values far below $m$ will be pulled further away from $m$, relative to that linear rescaling.
As a result, values at an equal distance, $d$ above and below $m$ before transformation will not be equally distant from it afterward - the transformed value above will be closer to $\log(m)$ than the transformed value below it will be. This would happen for every value of $d$.
So symmetric $X$ implies asymmetric $\log(X)$.
(I've been deliberately avoiding discussing the issue that all normal distributions put positive probability on the negative half-line, which makes for an immediate "no" but a less interesting answer.)
Now let's talk not about normality, but approximate normality. (For simplicity let's assume that the distribution is such that the values are going to be essentially always positive - i.e. if the original values were normal, the chance of a negative value is extremely low.)
There is one situation where approximately normal values tend to still be approximately normal after transformation.
That's when the standard deviation is very small compared to the mean (low coefficient of variation).
If you look at the above diagram, consider values on the x-axis in a very narrow band around $m$. The pulling-in/stretching-out effect is minimal (the black curve doesn't have room to move far away from the blue tangent line), and so the shape still looks normal.
Here's an example: the top plot is a set of approximately normal data (the Q-Q plot shows a fairly straight line), and its log is also approximately normal (the Q-Q plot still shows a fairly straight line). That's because the coefficient of variation in the original values was pretty small (somewhere around 0.2 I think) - the nonlinear transformation was still nearly linear in the narrow range of values around the middle.
In this situation, the delta method does indeed tend to be useful at giving approximate values for the mean and variance of the log-values, though it would not actually be the distribution of the log of an exactly normal random variate.
If $X$ is normally distributed, can $\log(X)$ also be normally distributed?
Other answers have stated that it is not possible. Indeed, theoretically, it is not possible. If $X$ is normally distributed, then $X$ can have negative values and $\log(X)$ does not exist for negative input.
In practice: Yes
However, in practice, one might deal with a distribution that is only approximately normal distributed, and which has a domain $X>0$. For such a distribution it might be still interesting to imagine what will happen when we take the transform $\log(X)$.
The Delta method
The delta method approximates the transformation $\log(X)$ by linearization. This approximation will work well when the difference around the mean (the point where you linearize) is not too large.
The image below illustrates the transform for different coefficient of variation of $X$ (See for similar images here).
You can see how the image on the left is not so close to a linear transform and the resulting histogram of $Y = \log(X)$, drawn in the margin, does not resemble so much a normal distribution and is skewed. However, the image on the right is closer to a linear transform and the transformed variable does resemble reasonably a normal distribution.
The values of $\mu$ and $\sigma$ in the last/right image show that the Delta method works for that case:
> ymean  2.995604 > log(xmean)  2.995921 > > ysig  0.02521176 > xsig/20  0.02519255 >
More precise than the Delta method
The Delta method will be less accurate when the $\sigma/\mu$ is larger because the approximation with linearization is less accurate.
The image below demonstrates this. It shows simulations of 10k points for $Y = \log(X)$ where $X \sim N(\mu_X = 1,\sigma_X = CV)$ where $CV$ has been varied (values $X<0$ were removed).
The red broken curve illustrates that the mean of $Y = \log(X)$ can be reasonably approximated by inverting the formula for the mean of a log-normal distributed variable.
If $Y = \exp(X)$ or $X = \log(Y)$, where $X \sim N(\mu_X,\sigma_X^2)$ then $\mu_Y = \exp(\mu_X + 0.5 \sigma_X^2) $, and the inverse $\mu_X \approx \log(\mu_Y)-0.5 \sigma_X^2$.
We could also do the same for the relatioship for the variance of a log-normal distribution but that is a bit awkward expression, so we simplify things a bit and fill in the Delta approximation $\sigma_X \approx \sigma_Y/\mu_Y$.
So we end up with
$$\mu_X \approx \log(\mu_Y)-0.5 \sigma_Y^2/\mu_Y^2$$
this is the red curve in the above plot and it seems to correspond well with the data.
In this question:
one deals with a logarithm of $X$ where it is the percent of monthly return of an investment. The mean is 1.01 and the sd = 0.04 such that the coefficient of variation is very small.
In that question, the delta method works, but the more precise method even better.