Peculiar Logarithms

$\begin{aligned} P & = \log_3 5 \times \log_5 7 \times \log_7 9 \times \cdots \times \log_{23} 25 \times \log_{25} 27 & \small \color{#3D99F6} \text{Convert to same base} \\ & = \frac {\log 5}{\log 3} \times \frac {\log 7}{\log 5} \times \frac {\log 9}{\log 7} \times \cdots \times \frac {\log 25}{\log 23} \times \frac {\log 27}{\log 25} \\ & = \frac {\cancel{\log 5}}{\log 3} \times \frac {\cancel{\log 7}}{\cancel{\log 5}} \times \frac {\cancel{\log 9}}{\cancel{\log 7}} \times \cdots \times \frac {\cancel{\log 25}}{\cancel{\log 23}} \times \frac {\log 27}{\cancel{\log 25}} \\ & = \frac {\log 27}{\log 3} = \frac {3\log 3}{\log 3} = \boxed 3 \end{aligned}$