How to interpret Standardized Coeff Beta when DV (in %) is transformed using log feature?

Hello everyone, I am analyzing the influence of some stores attributes (e.g., quality, price, in-store service, communication ...) on customer loyalty (more precisely share of wallet = how much a customer allocates to a specific store in %: 0-100).

I transformed the dependent variables (share of wallet) using the log feature in order to overcome non-normality of data.
However, I cannot figure out how to interpret the final number in an economic way.
Basic interpretation shows that increasing the IV by one-unit would increase the DV by 100*<Standardized Beta coeff>%.

Since the original unit of measurement was already in percentage (share of wallet), what would be the final interpretation?
Do I have to multiply 100*<Standardized Beta coeff>% by the average percentage allocated? or ...?
Also, my descriptive data are on an average level for each store, not per individual customer ...

Thanks a million for your help !!!



Less is more. Stay pure. Stay poor.
I am thinking it is just, for every standard deviation increase in the IV the DV increases blank (beta coefficient) percent. There may be some rule for getting a more exact outcome if beta coefficient is greater than 10%, so use a conversion, but you will have to look that up. There is a good example of conversions in the journal EPIDEMIOLOGY.