# Why is temperature defined like this?

I am reading Blundell and it's said here that if there are two systems with energies $E_1$ and $E_2$ in contact with each other and if $\Omega(E)$ represents the number of microstates for given energy $E$ then the two systems will be at equilibrium when the product $\Omega_1(E_1)\Omega_2(E_2)$ is maximised. So differentiating this with respect to either $E_1$ or $E_2$ and setting it to zero we get: $$\Omega_2(E_2)\frac{d\Omega_1(E_1)}{dE_1} + \Omega_1(E_1)\frac{d\Omega_2(E_2)}{dE_1} = 0$$ Since $dE_1 = -dE_2$ after some rearranging we get: $$\frac{d \ln\Omega_1}{dE_1} = \frac{d \ln\Omega_2}{dE_2}$$

Now it's said that since the systems are at equilibrium we can say both are at the same temperature T and we define temperature T as $$\frac{1}{k_B T}= \frac{d \ln\Omega}{dE}$$ where $k_B = 1.3807 \times 10^{-23} J K^{-1}$

My question is why exactly was the definition of temperature chosen in this way? Does equilibrium guarantee that both systems will arrive at the same temperature?