Can we revert law of cosines into its original complex identity?

Question

Write for $a$,$b$,$c$ positive and $z=x+iy$ complex:

$$a^z+b^z=c^z$$

This is equivalent to the system:

$$a^x\cos(y\ln(a))+b^x\cos(y\ln(b))=c^x\cos(y\ln(c))$$ $$a^x\sin(y\ln(a))+b^x\sin(y\ln(b))=c^x\sin(y\ln(c))$$