Solve for t.  ROUND YOUR FINAL ANSWER TO 2 DECIMAL 

PLACES.

5 e Superscript 2.2 t Baseline equals 508715e2.2t=50871


We are given the equation:

\[
5e^{2.2t} = 50871
\]

### Step 1: Isolate the exponential term.

To solve for \( t \), we first divide both sides of the equation by 5:

\[
e^{2.2t} = \frac{50871}{5}
\]

\[
e^{2.2t} = 10174.2
\]

### Step 2: Take the natural logarithm (ln) of both sides.

Since the natural logarithm is the inverse of the exponential function, we apply the natural logarithm to both sides:

\[
\ln(e^{2.2t}) = \ln(10174.2)
\]

Using the logarithmic property \( \ln(e^x) = x \), we simplify the left side:

\[
2.2t = \ln(10174.2)
\]

### Step 3: Solve for \( t \).

Now, calculate \( \ln(10174.2) \) and then divide by 2.2:

\[
t = \frac{\ln(10174.2)}{2.2}
\]

Let me compute the value.

\[
t = \frac{\ln(10174.2)}{2.2} \approx \frac{9.226}{2.2} \approx 4.19
\]

### Final Answer:
\[
t \approx 4.19
\]

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would you solve a similar equation if the base were not \( e \)?
2. What is the natural logarithm, and how does it differ from other logarithms?
3. Can you solve \( e^{3t} = 15000 \) using the same steps?
4. How would the equation change if we used logarithms with a different base (e.g., base 10)?
5. How can you graph the function \( 5e^{2.2t} \) to verify the solution visually?

**Tip**: Always isolate the exponential term first before applying logarithms when solving such equations.

Solve for t. ROUND YOUR FINAL ANSWER TO 2 DECIMAL PLACES. 5e^(2.2t) = 50871

Learn how to solve the exponential equation 5e^(2.2t) = 50871 with a detailed, step-by-step explanation. This problem involves isolating exponential terms, applying natural logarithms, and solving for t. Suitable for high school students in Grades 9-12.

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