Bu erda biz bakteriyalarning dastlabki soni 250 va shuning uchun \({Q_0} = 250\) ekanligini tan olishimiz bilan boshlashimiz mumkin. Shunday qilib, tenglama shunday bo'ladi:

\[Q\left( t \right) = 250{{\bf{e}}^{k\,t}}\]

Endi biz \(Q\left( 5 \right) = 1600\) va uni yuqoridagi tenglamaga kiritish shuni ham bilamizki,

\[1600 = Q\left( 5 \right) = 250{{\bf{e}}^{5k}}\]

dan texnikalardan foydalanishimiz mumkin Logarifm tenglamalarini yechish \(k\) qiymatini aniqlash uchun bo'lim.

\[\begin{align*}1600 & = 250{{\bf{e}}^{5k}}\\ \frac{{1600}}{{250}} & = {{\bf{e}}^{5k}}\\ \ln \left( {\frac{{32}}{5}} \right) & = 5k\\ k & = \frac{1}{5}\ln \left( {\frac{{32}}{5}} \right) = 0.3712596\end{align*}\]

Sizning afzalliklaringizga qarab, biz aniq qiymat yoki kasr qiymatidan foydalanishimiz mumkin. Shuni yodda tutingki, \(k\) eksponensial funktsiyaning ko'rsatkichida bo'lganligi sababli, agar biz haddan tashqari yaxlitlashimiz mumkin bo'lgan qiymatdagi potentsial katta farqlarga yo'l qo'ymaslik uchun bir nechta o'nli kasrlardan foydalanishimiz kerak.

Bularning barchasini jamlagan holda, ushbu populyatsiya uchun eksponensial o'sish tenglamasi:

\[\require{bbox} \bbox[2pt,border:1px solid black]{{Q = 250{{\bf{e}}^{\frac{1}{5}\ln \left( {\frac{{32}}{5}} \right)\,\,t}}}}\]

Bu erda bizdan haqiqatan ham so'ralayotgan narsa tenglamani yechishdir,

\[2000 = Q\left( t \right) = 250{{\bf{e}}^{\frac{1}{5}\ln \left( {\frac{{32}}{5}} \right)\,\,t}}\]

va biz dan bilamiz Logarifm tenglamalarini yechish buni qanday qilish kerak bo'limi. Mana bu qism uchun yechim ishi.

\[\begin{align*}\frac{{2000}}{{250}} & = {{\bf{e}}^{\frac{1}{5}\ln \left( {\frac{{32}}{5}} \right)\,\,t}}\\ \ln \left( 8 \right) & = \frac{1}{5}\ln \left( {\frac{{32}}{5}} \right)\,\,t\\ t & = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{5\ln \left( 8 \right)}}{{\ln \left( {\frac{{32}}{5}} \right)}} = 5.6010}}\end{align*}\]

Aholi soni 2000 kishiga yetishi uchun 5601 kun kerak bo'ladi.