Question

use the exponential growth model, ( a = a_0 e^{kt} ), to show that the time it takes a population to double (to grow from ( a_0 ) to ( 2a_0 )) is given by ( t=\frac{ln 2}{k} ).
to find the time it takes to grow the population from ( a_0 ) to ( 2a_0 ), substitute ( a = 2a_0 ).
( 2a_0 = a_0 e^{kt} ) substitute.
( 2 = e^{kt} ) divide both sides by ( a_0 ).
( ln 2=ln square ) take the natural logarithm on both sides.

Explanation:

Step1: Substitute \(A = 2A_0\) into the formula

Given the exponential - growth model \(A=A_0e^{kt}\), when \(A = 2A_0\), we have \(2A_0=A_0e^{kt}\).

Step2: Simplify the equation

Divide both sides of the equation \(2A_0 = A_0e^{kt}\) by \(A_0\) (since \(A_0
eq0\)). We get \(2 = e^{kt}\).

Step3: Take the natural logarithm of both sides

Take the natural logarithm of both sides of the equation \(2 = e^{kt}\). Using the property \(\ln(e^{x})=x\), if \(y = e^{x}\), then \(\ln(y)=x\). So, \(\ln(2)=\ln(e^{kt})\).
Since \(\ln(e^{kt}) = kt\) (by the property \(\ln(e^{u})=u\) where \(u = kt\)), we have \(kt=\ln(2)\).

Step4: Solve for \(t\)

Divide both sides of the equation \(kt=\ln(2)\) by \(k\) (assuming \(k
eq0\)). We get \(t=\frac{\ln(2)}{k}\).

Answer:

The time \(t\) it takes for the population to double is \(t = \frac{\ln(2)}{k}\)