Rewrite the function *P* = 154(0.9)*t* in the form *P* = *P*0*ekt*.

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.

Learn how to rewrite the exponential decay function P = 154(0.9)^t into the form P = P0 e^(kt). This problem covers exponential functions, natural logarithms, and exponential decay concepts. Suitable for advanced high school students.

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