Question

use the exponential growth model, $a = a_0e^{kt}$, to show that the time it takes a population to double

to find the time it takes to grow the population from $a_0$ to $2a_0$, substitute $a = 2a_0$.

$2a_0 = a_0e^{kt}$ \quad substitute.

$2 = e^{kt}$ \quad divide both sides by $a_0$.

$\ln 2 = \ln e^{kt}$ \quad take the natural logarithm on both sides.

$\ln 2 = kt$ \quad simplify.

$\dfrac{\ln 2}{\square} = t$ \quad isolate $t$.

Explanation:

Step1: Recall the equation

We have the equation \( \frac{\ln 2}{\square}=t \) and from the previous step \( \ln 2 = kt \).

Step2: Solve for the denominator

To isolate \( t \) from \( \ln 2 = kt \), we divide both sides by \( k \). So \( t=\frac{\ln 2}{k} \). Comparing with \( \frac{\ln 2}{\square}=t \), the denominator should be \( k \).

Answer:

\( k \)