Seems that no matter what is substituted for x, the log (x-10) is never able to equal the log (x-3) plus 3. Is this equation unsolvable, or are there theoretical/imaginary solutions?

2990/999 is a real number, but the logs are both complex numbers. So, you can only find the answer using algebra, or using a numeric environment that handles complex numbers.

I should have gone back and verified that the solution didn't break any rules, which it does.
But OP needs to specify whether we are talking strictly reals or not.

Given it's high school math it almost certainly is.