TeX source:

\begin{array}{ll}A=A_{0} e^{k t} & \text { The continuous growth formula. } \\0.5 A_{0}=A_{0} e^{k-5730} & \text { Substitute the half-life for } t \text { and } 0.5 A_{0} \text { for } f(t) . \\0.5=e^{5730 k} & \text { Divide by } A_{0} . \\\ln (0.5)=5730 k & \text { Take the natural log of both sides. } \\k=\frac{\ln (0.5)}{5730} & \text { Divide by the coefficient of } k . \\A=A_{0} e^{\left(\frac{\ln (0.5)}{5730}\right) t} & \text { Substitute for } k \text { in the continuous growth formula. }\end{array}