I have not taken statistics or calculus for 15 years, and I have been struggling to figure out how to create this curve for a project that I am working on. Any help will be appreciated.

I have a set of values scored from 1-100. I need a formula to transform these values into an s-curve. Scores above the mean need to take the form of a base-75 logarithmic curve; scores below the mean need to assume negative values that approximate a mirror image of the positive scores. The result will be a s-curve that sets the theoretical mean value of 50.5 to 0.

Thank you.