I have dependent monetary variable which is a lognormally distributed , i transformed it into log normally distributed using log function in R , after that mean and median has been very close. Now problem with one independent variable which has multimodal distribution. Do i need to convert this independent variable as log normally distributed as well? If yes how can i do this?
No. There is no requirement that independent variable being normal distributed.
Linear transformation (including scaling) on independent variable is meaningless. Non-linear transformation (including log) is needed if you find or know that there is non-linear relationship between dependent variable and independent variable.
Keep the variables with linear relationship as they are; make the suitable transformations on variables with non-linear relationship.