Answer
5
Work Step by Step
$$(\log_{2}3)\times(\log_{3}4)\times(\log_{4}5) ... \times(\log_{31}32)$$
$$\frac{\log3}{\log2}\times\frac{\log4}{\log3}\times\frac{\log5}{\log4}...\frac{\log31}{\log30}\times\frac{\log32}{\log31}$$
$$\frac{1}{\log2}\times\frac{\log3}{\log3}\times\frac{\log4}{\log4}...\frac{\log30}{\log30}\times\frac{\log31}{\log31}\times\log32$$
$$\frac{\log32}{\log2}$$
$$\log_232$$
$$5$$
