once #95%# of the kolesarium has decayed, #5%# of the original amount will be left.
a half-life is the amount of time that it takes for the mass of a substance to halve.
after #1# half-life, half of the original mass will be left.
after #2# half-lives, half of the mass from the first half-life will be left. this is half of half of the original mass - in other words, a quarter.
this is because #1/4# is #1/2 * 1/2#, or #(1/2)^2#.
after #1# half-life, #(1/2)^1# of the original mass is left.
after #2# half-lives, #(1/2)^2# of the original mass is left.
this shows that the mass after a certain number #n# of half lives is #(1/2)^n# of the original mass.
here, the mass after #n# half-lives is #5%#, or #0.05#, of the original mass.
if #(1/2)^n# of the original mass is #5%# of the original mass, then #(1/2)^n# is #0.05#.
if #(1/2)^n = 0.05#, then #n = log_0.5(0.05)#
#n = 4.322# (3d.p.)
this means it takes about #4.322# half-lives for the original mass to decrease by #95%#.
here, one half-life is #333# years.
#4.322..# half lives is about #1439# years (nearest whole number)
