Converting from log-standardized units to ordinary units - challenge with interpreting correctly

I have a variable X that I first log (base e)-transformed say log_X and then standardized i.e., log_X_s = (log_X - mean(log_X))/sd(log_X). I'm trying to use log_X_s in a logistic regression model and have the following output:

    Wald chi2(1)      =    7238.99
Log likelihood = -45419.208                     Prob > chi2       =     0.0000
-----------------------------------------------------------------------------------------
retained_p_day1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------------+----------------------------------------------------------------
log_X_s |   .7569809    .008897    85.08   0.000      .739543    .7744188
_cons |   .6084474   .0304769    19.96   0.000     .5487137     .668181
------------------------+----------------------------------------------------------------
nwikiproject            |
var(_cons)|   .9395261   .0492722                      .8477517    1.041236
-----------------------------------------------------------------------------------------
LR test vs. logistic model: chibar2(01) = 9298.74     Prob >= chibar2 = 0.0000


The distribution of log_X is shown below

Variable       n     Mean     S.D.      Min      .25      Mdn      .75      Max
-------------------------------------------------------------------------------
log_X        91882   6.78     3.35     0.00     4.50     7.42     9.42    13.85
-------------------------------------------------------------------------------


I'm trying to interpret the results back to the original scale and want to make sure I'm doing it correctly.

Interpretation 1:

A unit increase in log_X_s i.e., a 1 s.d. increase (i.e., 3.35 units) in log_X i.e., a 28.5 times (which is exp(3.35)) increase in X is associated with a exp(0.7569809) = 113% increase in the odds of retention.

So, a one time increase in X (i.e., doubling the quantity of X) is associated with a 113/28.5 ~ 4% increase in the odds of retention.

Interpretation 2:

An increase by 3.35 units on the log scale is associated with a 113% increase in the odds of retention. Therefore, a 1 unit increase on log scale ~ 113/3.35 is associated with 34% increase in the odds of retention. So, e times increase on the linear scale is associated with 34% increase in the odds of retention. Therefore, a 1x increase on the linear scale is associated with a 12.6% increase in the odds of retention.

Clearly, I'm getting confused with something somewhere and either one of these is wrong, or both. Could someone correct me where I'm going wrong with the interpretation? Any help is greatly appreciated!

• It's difficult to follow the logic of the second interpretation. Maybe you could resolve the confusion by re-running the regression on the unstandardized logarithms and comparing the coefficients. – whuber Aug 11 '17 at 23:01
• @whuber thanks for the input. Unfortunately, though, my model is not as simple as this. I have continuous by continuous interactions. Since I'm supposed to interpret simple effects, I feel it quite challenging to translate them back to original scale. I'm clueless if I'm doing them correctly. – rk567 Aug 12 '17 at 20:42
• I am suggesting there is a simple way to learn from your software--and it does not matter what your model is. Use any model. Use any data. Run the model on an unstandardized log variable and then on the standardized version and compare the coefficients. – whuber Aug 12 '17 at 20:44