Solve the equation. Give an exact​ solution, and also an approximate solution to four decimal places.
e Superscript 5 x Baseline equals 10
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Part 1
a. The exact solution is x equals

enter your response here.
Part 2
b. The approximate solution is x almost equals

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

To solve the equation \( e^{5x} = 10 \), follow these steps:

### Part 1: Exact Solution
1. Start by taking the natural logarithm of both sides to get rid of the exponent:
   \[
   \ln(e^{5x}) = \ln(10)
   \]
2. Using the property \( \ln(e^a) = a \), simplify the left side:
   \[
   5x = \ln(10)
   \]
3. Divide both sides by 5 to solve for \( x \):
   \[
   x = \frac{\ln(10)}{5}
   \]

So, the exact solution is:
\[
x = \frac{\ln(10)}{5}
\]

### Part 2: Approximate Solution
1. Calculate \( \ln(10) \approx 2.3026 \).
2. Divide by 5:
   \[
   x \approx \frac{2.3026}{5} \approx 0.4605
   \]

So, the approximate solution to four decimal places is:
\[
x \approx 0.4605
\]

Let me know if you'd like further clarification on any steps!

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**Further Questions:**
1. How does the natural logarithm function help in solving exponential equations?
2. What would the solution be if the equation were \( e^{5x} = 20 \)?
3. How can this method be applied to equations with other bases, like \( 2^{5x} = 10 \)?
4. Why is it important to keep as many decimal places as possible until the final rounding?
5. How would you verify this solution?

**Tip:** Remember, the natural logarithm \( \ln(x) \) is the inverse of \( e^x \), making it a powerful tool to solve equations with \( e \) as the base.

Solve the equation. Give an exact solution, and also an approximate solution to four decimal places. e^5x = 10

Learn how to solve the exponential equation e^5x = 10, finding both an exact and approximate solution. Understand key concepts like the natural logarithm and the exponential-logarithmic inverse relationship, with step-by-step guidance and a four-decimal place approximation.

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