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\begin{array}{ll}\frac{1}{2} A_{0}=A_{o} e^{k t} & \\\frac{1}{2}=e^{k t} & \text { Divide by } A_{0} . \\\ln \left(\frac{1}{2}\right)=k t & \text { Take the natural log. } \\-\ln (2)=k t & \text { Apply laws of logarithms. } \\-\frac{\ln (2)}{k}=t & \text { Divide by } k .\end{array}