The maximum population can be found by solving \(\displaystyle{P}'={0}\) for P while the fastest growth can be reached by equating the differentiation of the population rate by zero, then solving

(i.e) by solving \(\displaystyle{P}\text{}{0}\) for P as follows

\(\displaystyle{P}\text{}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left(\alpha{\ln{{\left({\frac{{{k}}}{{{P}}}}\right)}}}{P}\right)}={0}\)

\(\displaystyle\Rightarrow{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({\ln{{\left({\frac{{{k}}}{{{P}}}}\right)}}}{P}\right)}={0}\)

\(\displaystyle\Rightarrow{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({P}{\ln{{k}}}-{P}{\ln{{P}}}\right)}={0}\)

\(\displaystyle\Rightarrow{P}'{\ln{{k}}}-{P}'{\ln{{P}}}-\neg{\left\lbrace{P}\right\rbrace}{\frac{{{P}'}}{{\neg{\left\lbrace{P}\right\rbrace}}}}={0}\) (product rule and chain rule)

\(\displaystyle\Rightarrow{P}'{\ln{{k}}}-{P}'{\ln{{P}}}-{P}'={0}\)

\(\displaystyle\Rightarrow{\ln{{k}}}-{\ln{{P}}}-{1}={0}\) (dividing by P')

\(\displaystyle\Rightarrow{\ln{{P}}}={\ln{{k}}}-{1}\)

\(\displaystyle\Rightarrow{\ln{{P}}}={\ln{{k}}}-{\ln{{e}}}\) (logatithmic properties)

\(\displaystyle\Rightarrow{\ln{{P}}}={\ln{{\left({\frac{{{k}}}{{{e}}}}\right)}}}\) (logatithmic properties)

\(\displaystyle\Rightarrow{P}={\frac{{{k}}}{{{e}}}}\)

(i.e) by solving \(\displaystyle{P}\text{}{0}\) for P as follows

\(\displaystyle{P}\text{}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left(\alpha{\ln{{\left({\frac{{{k}}}{{{P}}}}\right)}}}{P}\right)}={0}\)

\(\displaystyle\Rightarrow{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({\ln{{\left({\frac{{{k}}}{{{P}}}}\right)}}}{P}\right)}={0}\)

\(\displaystyle\Rightarrow{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({P}{\ln{{k}}}-{P}{\ln{{P}}}\right)}={0}\)

\(\displaystyle\Rightarrow{P}'{\ln{{k}}}-{P}'{\ln{{P}}}-\neg{\left\lbrace{P}\right\rbrace}{\frac{{{P}'}}{{\neg{\left\lbrace{P}\right\rbrace}}}}={0}\) (product rule and chain rule)

\(\displaystyle\Rightarrow{P}'{\ln{{k}}}-{P}'{\ln{{P}}}-{P}'={0}\)

\(\displaystyle\Rightarrow{\ln{{k}}}-{\ln{{P}}}-{1}={0}\) (dividing by P')

\(\displaystyle\Rightarrow{\ln{{P}}}={\ln{{k}}}-{1}\)

\(\displaystyle\Rightarrow{\ln{{P}}}={\ln{{k}}}-{\ln{{e}}}\) (logatithmic properties)

\(\displaystyle\Rightarrow{\ln{{P}}}={\ln{{\left({\frac{{{k}}}{{{e}}}}\right)}}}\) (logatithmic properties)

\(\displaystyle\Rightarrow{P}={\frac{{{k}}}{{{e}}}}\)