Solve the equation to four decimal places.

e Superscript xexequals=5.3985.398

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Part 1

The solution is 

xalmost equals≈enter your response here. 

​(Round to four decimal​ places.)

To solve the equation \( e^x = 5.398 \), we need to isolate \( x \). This involves taking the natural logarithm of both sides. Here are the steps:

1. Start with the equation:
   \[ e^x = 5.398 \]

2. Take the natural logarithm (ln) of both sides:
   \[ \ln(e^x) = \ln(5.398) \]

3. Use the property of logarithms that \( \ln(e^x) = x \cdot \ln(e) \) and \( \ln(e) = 1 \):
   \[ x = \ln(5.398) \]

Now, we just need to calculate the natural logarithm of 5.398 to four decimal places.

\[ x \approx \ln(5.398) \]The solution to the equation \( e^x = 5.398 \) is \( x \approx 1.6860 \) when rounded to four decimal places.

Would you like more details or have any questions? Here are 8 related questions to expand on this information:

1. What is the base \( e \) in natural logarithms?
2. How can you verify the solution by substitution?
3. What is the importance of the natural logarithm in mathematics?
4. How does changing the value of \( 5.398 \) affect the solution?
5. What are some real-world applications of exponential equations?
6. How would you solve the equation \( e^{2x} = 5.398 \)?
7. What is the difference between natural logarithms and common logarithms?
8. How can you use logarithms to solve equations involving other bases?

**Tip:** Remember that natural logarithms are useful for solving equations involving exponential growth or decay.

Learn how to solve the equation \( e^x = 5.398 \) step-by-step to four decimal places. This problem involves exponential functions and natural logarithms, crucial concepts in advanced mathematics. Suitable for high school students preparing for calculus or college-level mathematics courses.

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