Question

use the exponential growth model, $a = a_0e^{kt}$, 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 = $

Explanation:

Step1: Substitute values into formula

Substitute $A = 2A_0$ into $A=A_0e^{kt}$, we get $2A_0=A_0e^{kt}$.

Step2: Simplify the equation

Divide both sides of $2A_0 = A_0e^{kt}$ by $A_0$ (assuming $A_0
eq0$), we have $2 = e^{kt}$.

Step3: Take natural - logarithm of both sides

$\ln(2)=\ln(e^{kt})$.

Step4: Use logarithm property

Since $\ln(e^{x}) = x$, then $\ln(2)=kt$.

Step5: Solve for $t$

Divide both sides by $k$ to get $t=\frac{\ln(2)}{k}$.

Answer:

$t=\frac{\ln(2)}{k}$