Answer :
The population growth formula is :
[tex]P=P_oe^{rt}[/tex]where P = total population after time t
Po = initial population
e = euler's number
r = percent rate of growth
t = time in years
From the problem, the population doubles in 15 years, so we have :
P = 2Po
t = 15
Using the formula above :
[tex]\begin{gathered} 2P_o=P_oe^{15r}_{} \\ 2=e^{15r} \end{gathered}[/tex]Take ln of both sides :
[tex]\ln 2=\ln (e^{15r})[/tex]Note that :
[tex]\begin{gathered} \ln a^m=m\ln a \\ \text{and} \\ \ln e=1 \end{gathered}[/tex]The natural logarithm of a raised to m is the same as m multiplied by the natural logarithm of a.
So the equation will be :
[tex]\begin{gathered} \ln 2=15r\ln e \\ \ln 2=15r \\ r=\frac{\ln 2}{15} \\ r=0.0462 \end{gathered}[/tex]The answer is 0.0462 or 4.62%