Compute log(1/2) + log(2/3) + log(3/4) + .... + log(99/100). (All logarithms are in base 10.)
log(1/2) + log(2/3) + log(3/4) + .... + log(99/100)
Note that we can write
log (1/2 * 2/3 * 3/4 * 4/5 * ....... * 98 / 99 * 99 / 100) =
log ( 1 * 2/2 * 3/3 * 4/4 * 5/5 * ........* 98/98 * 99/99 * 1 /100) =
log ( 1 * 1 * 1 * 1 * 1 * ........ * 1 * 1 * 1/ 100) =
log (1/100) =
log 10^(-2) =
-2 log 10 =
-2