CMB temperature

If the ionisation temperature of hydrogen is \(T=10^4\)K, the CMB decoupled at a redshift of \(z=10^3\), and if for photons the temperature drops proportional to \(1/a\), shouldn’t the CMB have a temperature of \(10\)K instead of \(3\)K?

bonus question: Would it be possible to generalise the hyperbolic functions to a basis \(a\neq e=\exp(1)\) in a way that \(\mathrm{asinh}(x)\) behaves asymptotically as \(\log_{a}(x)\) for \(x\gg 1\)?