To rewrite the function $P = 154(0.9)^t$ in the form $P = P_0 e^{kt}$, we need to express the base $0.9$ in exponential form.

Step 1: Identify the initial population $P_0$.

The initial population $P_0$ is the value of $P$ when $t = 0$. In the given equation, $P_0 = 154$.

Step 2: Rewrite $0.9$ as an exponential function.

The function can be rewritten using the property $a^t = e^{t \ln(a)}$:

$0.9^t = e^{t \ln(0.9)}$

Step 3: Substitute into the original equation.

Now, replace $0.9^t$ with $e^{t \ln(0.9)}$:

$P = 154 \cdot e^{t \ln(0.9)}$

Step 4: Identify $k$.

Here, $k$ is $\ln(0.9)$, which is a negative value:

$k = \ln(0.9) \approx -0.10536$

So the equation in the desired form is:

$P = 154 e^{-0.10536t}$

Would you like more details or have any questions about this?

Here are 5 related questions to consider:

1. How would the equation change if the initial population $P_0$ was different?
2. What does the negative sign in $k$ indicate about the population over time?
3. How would you find the time $t$ at which the population halves?
4. If the base was $1.1$ instead of $0.9$, what would the continuous growth rate $k$ be?
5. How does the continuous decay rate $k$ compare to the nominal decay rate?

Tip: When dealing with exponential decay or growth, converting to the continuous form $P = P_0 e^{kt}$ can simplify solving problems involving continuous rates of change.