One of the predictors in my logistic model has been log transformed. How do you interpret the estimated coefficient of the log transformed predictor and how do you calculate the impact of that predictor on the odds ratio?
If you exponentiate the estimated coefficient, you'll get an odds ratio associated with a $b$-fold increase in the predictor, where $b$ is the base of the logarithm you used when log-transforming the predictor.
I usually choose to take logarithms to base 2 in this situation, so I can interpet the exponentiated coefficient as an odds ratio associated with a doubling of the predictor.
@gung is completely correct, but, in case you do decide to keep it, you can interpret the coefficient has having an effect on each multiple of the IV, rather than each addition of the IV.
One IV that often should be transformed is income. If you included it untransformed, then each (say) \$1,000 increase in income would have an effect on the odds ratio as specified by the odds ratio. On the other hand, if you took log(10) of income, then each 10 fold increase in income would have the effect on the odds ratio specified in the odds ratio.
It makes sense to do this for income because, in many ways, an increase of \$1,000 in income is much bigger for someone who makes \$10,000 per year than someone who makes \$100,000.
One final note - although logistic regression makes no normality assumptions, even OLS regression doesn't make assumptions about the variables, it makes assumptions about the error, as estimated by the residuals.