The equation is:

\[
4e^{2.1t} = 38423
\]

To solve for \( t \), follow these steps:

### Step 1: Divide both sides by 4 to isolate the exponential term:

\[
e^{2.1t} = \frac{38423}{4} = 9605.75
\]

### Step 2: Take the natural logarithm (ln) of both sides to eliminate the exponential:

\[
\ln(e^{2.1t}) = \ln(9605.75)
\]

Since \(\ln(e^x) = x\), we have:

\[
2.1t = \ln(9605.75)
\]

### Step 3: Calculate the natural logarithm of 9605.75:

\[
\ln(9605.75) \approx 9.170
\]

### Step 4: Solve for \( t \):

\[
t = \frac{9.170}{2.1} \approx 4.367
\]

Thus, \( t \approx 4.367 \).

Would you like a detailed breakdown of these steps or have any questions?

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Here are 5 related questions to consider:
1. How do you solve for \( t \) if the base of the exponential was not \( e \)?
2. What is the significance of the natural logarithm in solving exponential equations?
3. How would the solution change if the coefficient of \( t \) were negative?
4. How does this method apply to solving other exponential growth problems?
5. Can you solve \( e^{kt} = A \) for any general \( k \) and \( A \)?

**Tip:** When solving exponential equations, taking the logarithm of both sides is key to simplifying.

Learn how to solve the exponential equation 4e^2.1t = 38423 using logarithms. This step-by-step guide explains the use of natural logarithms and exponential functions to isolate t. Suitable for high school students studying algebra and pre-calculus.

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