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To solve for \( t \) in the equation:

\[
7e^{7t} = 100
\]

follow these steps:

1. **Isolate the exponential term**:
   \[
   e^{7t} = \frac{100}{7}
   \]

2. **Take the natural logarithm of both sides** to remove the exponential:
   \[
   \ln(e^{7t}) = \ln\left(\frac{100}{7}\right)
   \]

3. **Simplify using the property \(\ln(e^x) = x\)**:
   \[
   7t = \ln\left(\frac{100}{7}\right)
   \]

4. **Solve for \( t \)** by dividing both sides by 7:
   \[
   t = \frac{\ln\left(\frac{100}{7}\right)}{7}
   \]

### Final Answer:
\[
t = \frac{\ln\left(\frac{100}{7}\right)}{7}
\]

To get a numerical approximation:
\[
t \approx \frac{\ln(14.2857)}{7} \approx \frac{2.6593}{7} \approx 0.3799
\]

So, \( t \approx 0.3799 \).

Would you like further details or have any questions?

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Here are some related questions to deepen your understanding:

1. How do logarithmic properties help in solving equations involving exponents?
2. What is the significance of using the natural logarithm \(\ln\) in equations with base \(e\)?
3. How would the approach change if the equation was \(7e^{5t} = 100\)?
4. How can you solve an equation of the form \( a \cdot e^{bx} = c \) in general?
5. What is the purpose of isolating terms with the unknown variable before applying logarithms?

**Tip**: Always remember to simplify terms step-by-step, especially in exponential and logarithmic equations, as it helps in maintaining accuracy throughout the solution process.

Solve for the unknown, t, in the equation 7e^(7t) = 100.

Learn how to solve the exponential equation 7e^(7t) = 100 to find the value of t. This solution involves isolating the exponential term and applying natural logarithms, providing a clear step-by-step approach. Ideal for high school students in Grades 10-12.

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