An "exponential" can be written with any base: if \(\displaystyle f(x)= a^x\) then \(\displaystyle f(x)= b^{log_b(a^x)}= b^{xlog_b(a)}\) so it's just a different coefficient. The "e" base is often used because it has nice "Calculus" properties which are not relevant here.
Because it says that "After 150 thousand years, only 1/8 of the original amount of a particular radioactive waste will remain" you can write \(\displaystyle P(t)= P(0)(1/8)^{t/150000}\). But \(\displaystyle 8= 2^3\), so this can be written as \(\displaystyle P(t)= P(0)(1/2^3)^{t/150000}= P(0)/2^{3t/150000}= P(0)/2^{t/50000}\). That is, because \(\displaystyle 8= 2^3\), the "half life" is 150000/3= 50000 years.
