Solve the equation. Give an exact​ solution, and also an approximate solution to four decimal places.
e Superscript 9 x Baseline equals 6
Question content area bottom
Part 1
a. The exact solution is xequals
  
StartFraction ln left parenthesis 6 right parenthesis Over 9 EndFraction
.
Part 2
b. The approximate solution is xalmost equals
  
enter your response here.

To solve the equation \( e^{9x} = 6 \), we proceed as follows:

### Part 1: Exact Solution
Starting from the equation:
\[
e^{9x} = 6
\]
Take the natural logarithm on both sides:
\[
\ln(e^{9x}) = \ln(6)
\]
Using the property \( \ln(e^a) = a \), we simplify to:
\[
9x = \ln(6)
\]
Now, solve for \( x \):
\[
x = \frac{\ln(6)}{9}
\]

Thus, the exact solution is:
\[
x = \frac{\ln(6)}{9}
\]

### Part 2: Approximate Solution
To find an approximate value, calculate \( \frac{\ln(6)}{9} \) to four decimal places.

\[
x \approx \frac{\ln(6)}{9} \approx 0.2986
\]

### Final Answers
- **Exact solution**: \( x = \frac{\ln(6)}{9} \)
- **Approximate solution**: \( x \approx 0.2986 \)

Would you like further details on this solution? Here are five related questions to deepen understanding:

1. How can we solve equations involving \( e^{kx} = b \) for other values of \( k \) and \( b \)?
2. What properties of logarithms are useful in simplifying exponential equations?
3. Why do we use the natural logarithm instead of other bases here?
4. How would the solution change if the equation were \( e^{9x} = 1 \) instead?
5. How can you verify the solution by substituting it back into the original equation?

**Tip:** Remember, taking the logarithm of both sides is a common approach to solving exponential equations!

Solve the equation. Give an exact​ solution, and also an approximate solution to four decimal places. e^(9x) = 6.

Learn how to solve the exponential equation e^(9x) = 6, with both exact and approximate solutions provided. This detailed solution demonstrates how to apply logarithmic properties to find x = ln(6) / 9 and an approximate value of x ≈ 0.2986. Ideal for students in Grades 10-12, covering key concepts like exponential equations and logarithms.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation