\(\therefore \,\frac{{e\frac{{ - 2000}}{T}}}{{e\frac{{ - 1000}}{T}}}= \frac{{{{10}^{15}}}}{{{{10}^{16}}}}\)
\(\therefore \,e\frac{{ - 1000}}{T}\, = \,{10^{ - 1}}\)
\(\therefore \,\ln \,{10^{ - 1}}\, = \,\frac{{ - 1000}}{T}\)
\(\therefore \,2.303\,\log \,{10^{ - 1}}\, = \,\frac{{ - 1000}}{T}\)
\(\therefore \,- 2.303 = \frac{{ - 1000}}{T}\)
\(\therefore \,2.303 = \frac{{1000}}{T}\)
\(\therefore \,T = \frac{{1000}}{{2.303}}\)