It is possible that there is an error. I do not have your data so I cannot confirm your observation.
I used weight and height in Big Class data table to fit a Exponential 2P model for each age. I compared the estimates:
I applied the Log() function to the weight and repeated my analysis:
The transformation changed the scale and shape as expected. My simple trial does not seem to replicate the problem that you see.
Can you show us the same plot as below for the fits to your data using the original response and the transformed response?