Formula for rate constant for the first order reaction My lecturer mentioned that the formula for the rate constant $k$ for the first order reaction is
$$k = \frac{2.0303}{t_{1/2}}\,\log\frac{[\ce{A}]_0}{[\ce{A}]_{t_{1/2}} - [\ce{A}]_0},\tag{1}$$
where $t_{1/2}$ is the half-life; $[\ce{A}]_0$ is the initial concentration; $[\ce{A}]_{t_{1/2}}$ is the concentration at half-life.
In which what I know recently is that the rate constant for the first order reaction is
$$
\begin{align}
&amp;  &amp; \ln\frac{[\ce{A}]_t}{[\ce{A}]_0} &amp;= -kt \tag{2}\\
&amp;\implies &amp; [\ce{A}]_t &amp;= [\ce{A}]_0\,\mathrm e^{-kt}\tag{3}
\end{align}
$$
I ask my teacher, but he said I should go and do more research, but I couldn't find it.
I am attaching a written answer as typing the log and other exponents is hard :(