Trouble finding inverse of y = 10^x / (10^x + 1)

This is surely where a teacher is handy to have. Luckily, I also have this forum!

I'm obviously missing something, but I can't figure out the inverse of:

I can isolate one x variable, but can't figure out how to isolate it entirely.

I've tried a bunch of ways, but the main one is:

Take the log base 10 of both sides:

Split off the division into subtraction:

Then I can isolate one x:

I just can't seem to see the way forward. Please, someone help put me out of my misery. Would be much appreciated!