Deriving the buffer formula
$$\ce{pH} = \mathrm{p}K_\mathrm {a} + \log \left(\frac{\alpha}{1-\alpha} \right)$$
Multiply the equation by $-1$:

$$-\ce{pH} = -\mathrm{p}K_\mathrm {a} + \log \left(\frac{1-\alpha} {\alpha}\right)$$
Rearrange the equation
$$-\ce{pH} +\mathrm{p}K_\mathrm {a} = \log \left(\frac{1-\alpha} {\alpha}\right)$$

Apply the definition of $\log$

$$10^{-\ce{pH} +\mathrm{p}K_\mathrm {a}} = \frac{1-\alpha} {\alpha}=  \frac{1}{\alpha} -1$$
Rearrange the equation
$$1+ 10^{-\ce{pH} +\mathrm{p}K_\mathrm {a}} =  \frac{1}{\alpha} $$
So,
$$\alpha= \frac{1}{ 1+ 10^{-\ce{pH} +\mathrm{p}K_\mathrm {a}}  } $$