Question

Evaluate: ${\log }_3(1+\frac{1}{3})+{\log }_3(1+\frac{1}{4})+.......+{\log }_3(1+\frac{1}{80})$

Answer 1

${log}_3(1+\frac{1}{3})+{log}_3(1+\frac{1}{4})+{log}_3(1+\frac{1}{5}).....+{log}_3(1+\frac{1}{79})+{log}_3(1+\frac{1}{80})\Rightarrow {log}_3\left(\frac{4}{3}\right)+{log}_3\left(\frac{5}{4}\right)+{log}_3\left(\frac{6}{5}\right).....+{log}_3\left(\frac{80}{79}\right)+{log}_3\left(\frac{81}{80}\right)\Rightarrow {log}_3(\frac{4}{3}\times \frac{5}{4}\times \frac{6}{5}....\frac{80}{79}\times \frac{81}{80})\Rightarrow {log}_3(\frac{1}{3}\times \frac{81}{1})\Rightarrow {log}_3\left(27\right)\Rightarrow {log}_3\left(3^3\right)\Rightarrow 3{log}_3\left(3\right)=3\times 1=3$