Use the fact that the world population was $2560$ million in $1950$ and $3040$ million in $1960$ to model the population of the world in the second half of the $20$th century. (Assume that the growth rate is proportional to the population size.) What is the relative growth rate? Use the model to estimate the world population in $1993$ and to predict the population in year $2020$.
Solution We measure the time $t$ in years and let $t=0$ in the year $1950$. We measure the population $P(t)$ in millions of people.
Then $P(0)=2560$
and
$P(0)=3040$
$P(t)=P(0)e^{kt}=2560e^{kt}$
$P(10)=2560e^{kt}=3040$
This is where I have a problem.
I apply the natural logarithm to both sides of the equation.
$\ln 2560e^{10k}=\ln3040$
I move the exponent up front. I am not sure if I am allowed to move the constant and variable.
$10k\ \ln 2560e = \ln 3040$
I am trying to isolate $k$. How can I do that? This is more an algebra problem at this point.
Best Answer
$\ln(2560 e^{10k})=\ln (2560) + \ln(e^{10k})=\ln (2560) +10k \ln(e)=\ln (2560) + 10k$ I think you confused yourself with whether the base of the power was $e$ or $2560e$