Fun with Logarithms

$\begin{aligned} S & = \log_2 \left(1+\frac 12\right) + \log_2 \left(1+\frac 13\right) + \log_2 \left(1+\frac 13\right) + ... + \log_2 \left(1+\frac 1{31}\right) \\ & = \log_2 \frac 32 + \log_2 \frac 43 + \log_2 \frac 54 + ... + \log_2 \frac {32}{31} \\ & = \log_2 \left(\frac 32 \times \frac 43 \times \frac 54 \times ... \times \frac {32}{31} \right) \\ & = \log_2 \left(\frac {\cancel 3}2 \times \frac {\cancel 4}{\cancel 3} \times \frac {\cancel 5}{\cancel 4} \times ... \times \frac {32}{\cancel{31}} \right) \\ & = \log_2 \frac {32}2 = \log_2 16 = \boxed{4} \end{aligned}$

$\log_{2}(\frac{3}{2})+\log_{2}(\frac{4}{3})+.....\log_{2}(\frac{32}{31})\\ \text{According to properties of logarithms the above expression is equal to,}\\ \log_{2}(\frac{3}{2}×\frac{4}{3}×.....×\frac{32}{31})\\ \log_{2}(\frac{32}{2})\\ \log_{2}16=\boxed{4}$