What is the general formula for exponential growth of a population?

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Answer 1 · 2018-03-02T20:02:22

Population $[P]= Ce^[kt$

If the rate of growth $P$ is proportional to itself, then with respect to time $t$ ,

$[dP]/dt=kP$ , ....inverting both sides, ..... $dt/[dP]=[1]/[kP$ and so integrating both sides

$intdt=int[dP]/[kP$ , thus,..... $t=1/klnP +$ a constant............ $[1]$

Suppose $P$ is some value $C$ when $t=0$ , substituting

$0=1/klnC+$ constant, therefore the constant $= -1/klnC$ and so substituting this value for the constant in ... $[1]$ we have ,

$t= 1/k[ln P-lnC]$ = $1/k ln[P/C]$ , therefore , $kt=ln[p/C]$ [ theory of logs] and so

$e^[kt]=P/C$ ......giving $P=Ce^[kt$ . The constant $k$ will represent the excess of births over deaths or vice versa for a decreasing rate.