I agree with victor in that the use of standardized coefficients in linear regression is an acceptable way to tell relative impact (it is the primary reason standardized coefficients were created). This is more complex in logistic regression because there is no agreement on how to standardize coefficients and many different ones exist that won't lead to the same result commonly.

This is from Michigan State, which has an excellent reputation for stats.

If two independent variables are measured in exactly

the same units, we can asses their relative importance

in their effect on y quite simply

– The larger the coefficient, the stronger the effect

• Often, however, our explanatory variables are not all

measured in the same units, making it difficult to

assess relative importance

• This problem can be overcome for quantitative

variables by using standardized variables

http://polisci.msu.edu/jacoby/icpsr/regress3/lectures/week2/8.RelImport.pdf
Note that this author, although not all authors, argues that it is not a good idea to standardize dummy variables (because speaking of a one unit change makes limited sense when the unit can only change one unit).