The given equation is:

\[
5 \cdot e^{2x} = 25
\]

### Steps to solve for \( x \):

1. **Isolate the exponential term**:  
   Divide both sides of the equation by 5:

   \[
   e^{2x} = \frac{25}{5} = 5
   \]

2. **Take the natural logarithm (ln) of both sides**:  
   Apply the natural logarithm to both sides to get rid of the exponential:

   \[
   \ln(e^{2x}) = \ln(5)
   \]

3. **Simplify the logarithmic expression**:  
   Using the property \( \ln(e^y) = y \), we simplify the left-hand side:

   \[
   2x = \ln(5)
   \]

4. **Solve for \( x \)**:  
   Finally, divide both sides by 2:

   \[
   x = \frac{\ln(5)}{2}
   \]

Thus, the solution is:

\[
x = \frac{\ln(5)}{2}
\]

If you'd like, I can also provide the approximate decimal value of \( \ln(5) \).

Do you have any questions or need further details?

### Related questions:
1. How do you solve equations with different bases of exponents?
2. What is the natural logarithm and how is it used?
3. How do you simplify expressions involving \( e \) and logarithms?
4. What are the properties of logarithms that are useful in solving equations?
5. How can exponential functions model real-world phenomena?

**Tip:** Remember that logarithms are the inverse operations of exponentials!

Learn how to solve the exponential equation 5⋅e^{2x}=25 with step-by-step instructions. This solution involves isolating the exponential, applying the natural logarithm, and solving for x. Suitable for students in Grades 10-12, this problem covers key concepts like logarithms and exponential functions.

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