Octave code

syms P0 k t a.\(\displaystyle{P}={k}\cdot{\left({P}\frac{{0}}{{k}}\right)}^{{{e}^{{-{a}\cdot{t}}}}}.{v}{p}{a}{\left({s}{o}{l}{v}{e}{\left(\subset{s}{\left({P},{\left[{P}{0}{k}{a}\right]},{\left[{1}{10}^{{7}}{0.1}\right]}\right)}={5}\cdot{10}^{{6}},{t}\right)}\right)}\)

The result is

31.464555148849334319210049495334