This Compression Equation: floor(n) = n, floor(log(|x|)/log(10))

[Narret]

So I was playing around on a white board trying to make an equation that compresses information and ran into a bit of an issue... It has reached a level of math I don't know yet (pre-calc is my limit)... Im not sure if there is a right answer to this equation but Im going to post it here if someone is up to the challenge.

Solve for n

. . . . .$\displaystyle \normalsize{\left\{\lfloor n \rfloor \, =\, n,\, \left\lfloor \frac{\log(|x|)}{\log(10)}\right\rfloor\, >\, \left\lfloor \frac{\log(|n|)}{\log(10)} \right\rfloor,\, x\, =\, 1234,\, nx\, =\, n^{\left\lfloor \frac{\log(|x|)}{\log(10)} \right\rfloor + 1}\, -\, n\, +\, x\, \huge{\wedge}\normalsize\, n\, \neq\, 1\right\}}$

btw I don't know if there is any integer answer for n, the non integer constrained answer is ~10... But it may exist I don't know!
Also this equation would be better if you could have different numbers for each instance of n but their conjugate must still follow the same rules.