How to interpret descriptive statistics on log transformed variable I'm sitting with a question where we are asked to log transform a variable and then interpret the descriptive statistics on the log transformed variable. The variable represents the number of patents firms have.
I do not understand how to interpret results for this log transformed variable. To my understanding there is not much value in interpreting e.g. the mean.

 A: You used Stata and I will too, but absolutely nothing hinges on that. The principles cut across software.
The minimum and maximum are easier to think about the original scale, but if you were only given the results in the question, you could recover the original extremes just by exponentiation.
. di exp(1.791759)
5.9999972

. di exp(4.41884)
82.99995

Evidently the extremes were 6 and 83.
The mean  of the logarithms when exponentiated is the geometric mean, which is often a useful measure for positively skewed distributions with entirely positive values. Here it is about 21. The geometric mean is given by exp(mean(ln()).
. di exp(3.036484)
20.831869

It is worth noting that the geometric mean is close to halfway between the minimum and maximum on the logarithmic scale, which is consistent with an idea that taking the logarithms should have made the distribution more nearly symmetric, although naturally graphs should always be looked at too.
. di exp((1.791759 + 4.41884)/2)
22.315902

We will simulate a lognormal with the same summary measures.

. clear

. set obs 1482
Number of observations (_N) was 0, now 1,482.

. set seed 2803

. gen simulated = exp(rnormal(3.036484, 0.4614554))

. su simulated

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
   simulated |      1,482    23.20229    11.14353   4.538362   76.63311


It is not an accident that the SD on the log scale is close to the coefficient of variation on the original scale.
. di r(sd)/r(mean)
0.48027712

Here the common convention of reporting CV as a percent should be ignored.
A: The log transformation affects the calculation of statistics, but you can always transform them back to the original scale for interpretation.
Obviously the mean will be different depending on whether you calculated it on the original scale versus the log scale followed by unlogging, but it may be more meaningful for certain data than using the original scale.
You can also do things like calculate the confidence interval using the mean and std dev of the logged data, then unlog to get a more interpretable range on the original scale.
In the case of patents, I suspect the distribution is highly right skewed. The mean on the original scale might be a fairly large number that isn't at all representative of how many patents most firms actually have. You can also compare the confidence intervals for the mean, using both the original scale and the log-transformed numbers (which should be something like [8.43, 51.46] for a 95% CI), which will likely overlap your data distribution very differently and give you a better intuition on the matter.
A: If you use decimal log rather than natural, you can interpret it as a sort of "number of digits minus one". You can transform it to decimal logarithms if you divide by $\ln(10)$ (about $2.3025$).
Other than that, you may want to revert your data to its original scale for descriptive statistics that are hard to interpret in this way.
