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:

$\displaystyle y = \frac{10^x}{10^x + 1}$

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:

$\displaystyle log_{10}(y) = log_{10}(\frac{10^x}{10^x + 1})$

Split off the division into subtraction:

$\displaystyle log_{10}(y) = log_{10}(10^x) - log_{10}(10^x + 1)$

Then I can isolate one x:

$\displaystyle log_{10}(y) = x - log_{10}(10^x + 1)$

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