Solve the equation. Give an exact​ solution, and also an approximate solution to four decimal places.

e Superscript 2 x Baseline equals 2e2x=2

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Part 1

**a.** The exact solution is 

xequals=enter your response here.

Part 2

**b.** The approximate solution is 

xalmost equals≈enter your response here.

​(Do not round until the final answer. Then round to four decimal places as​ needed.)

To solve the equation \( e^{2x} = 2 \), we'll isolate \( x \).

### Part 1: Exact Solution
1. Start with the equation:
   \[
   e^{2x} = 2
   \]

2. Take the natural logarithm of both sides to solve for \( x \):
   \[
   \ln(e^{2x}) = \ln(2)
   \]

3. Apply the property \( \ln(e^a) = a \) on the left side:
   \[
   2x = \ln(2)
   \]

4. Divide by 2 to isolate \( x \):
   \[
   x = \frac{\ln(2)}{2}
   \]

So, the exact solution is:
\[
x = \frac{\ln(2)}{2}
\]

### Part 2: Approximate Solution
1. Now, calculate \( x \) approximately by substituting the value of \( \ln(2) \approx 0.693147 \):
   \[
   x \approx \frac{0.693147}{2} \approx 0.3466
   \]

So, the approximate solution to four decimal places is:
\[
x \approx 0.3466
\]

### Final Answer
**a.** The exact solution is \( x = \frac{\ln(2)}{2} \).

**b.** The approximate solution is \( x \approx 0.3466 \).

Would you like more details on any part? 

---

1. How do we know when to use natural logarithms in equations with \( e \)?
2. What properties of logarithms are used to simplify \( e^{2x} = 2 \)?
3. How can we check if the exact and approximate solutions are correct?
4. What are alternative methods to approximate \( x \) to four decimal places?
5. How does solving exponential equations differ from solving polynomial equations?

**Tip:** In exponential equations, taking the logarithm helps simplify expressions where the variable is an exponent.

Solve the equation. Give an exact solution, and also an approximate solution to four decimal places.

e^(2x) = 2

Part 1

a. The exact solution is x = enter your response here.

Part 2

b. The approximate solution is x ≈ enter your response here.

(Do not round until the final answer. Then round to four decimal places as needed.)

Learn how to solve the exponential equation e^(2x) = 2 with a step-by-step guide. This solution covers the use of natural logarithms to find the exact solution x = ln(2)/2 and its approximate value x ≈ 0.3466 to four decimal places. Suitable for high school students in Grades 10-12.

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