Answer
$5$
Work Step by Step
$$(\log_{2}3)\times(\log_{3}4)\times(\log_{4}5) ... \times(\log_{31}32)$$
$$\frac{\ln3}{\ln2}\times\frac{\ln4}{\ln3}\times\frac{\ln5}{\ln4}...\frac{\ln31}{\ln30}\times\frac{\ln32}{\ln31}$$
$$\frac{1}{\ln2}\times\frac{\ln3}{\ln3}\times\frac{\ln4}{\ln4}...\frac{\ln30}{\ln30}\times\frac{\ln31}{\ln31}\times\ln32$$
$$\frac{\ln32}{\ln2}$$
$$\log_232$$
$$\log_2 2^5$$
$$5$$
