What is half life (t_1/2) of a reaction? Derive the equation for the half life value of zero and first order reaction

The time required for completion of half of the concentration of reactant is known as half life (t_1/2)
For first order $\mathrm{k}=\frac{2.303}{\mathrm{t}}\log \frac{\mathrm{a}}{(\mathrm{a}-\mathrm{x})}$
$\mathrm{k}=\frac{2.303}{{\mathrm{t}}_{1/2}}\log \frac{\mathrm{a}}{\mathrm{a}/2}$
a = initial concentration of reactant
$\begin{array}{l}=\frac{2.303}{{\mathrm{t}}_{1/2}}{\log }_{10}^2\text{ if }\mathrm{t}={\mathrm{t}}_{1/2},\mathrm{a}-\mathrm{x}=\mathrm{a}/2\\ =\frac{2.303}{{\mathrm{t}}_{1/2}}\times 0.3010\mathrm{k}=\frac{0.693}{{\mathrm{t}}_{1/2}}\end{array}$
For zero order $\mathrm{k}=\frac{\mathrm{x}}{\mathrm{t}}$
Since if $\mathrm{t}={\mathrm{t}}_{1/2},\mathrm{x}=\mathrm{a}/2$
$\mathrm{k}=\frac{\mathrm{a}}{2{\mathrm{xt}}_{1/2}},\mathrm{k}=\frac{\mathrm{a}}{2\times {\mathrm{t}}_{1/2}},{\mathrm{t}}_{1/2}=\frac{\mathrm{a}}{2\mathrm{k}}$