https://file.aiquickdraw.com/mathbot/file/1730521291921dra4bpjg.jpg

The question asks to solve for \( x \) given the equation:

\[
e^{7x} = 27
\]

To solve for \( x \), we can take the natural logarithm of both sides to isolate \( x \):

1. Start by applying the natural logarithm (ln) to both sides:
   \[
   \ln(e^{7x}) = \ln(27)
   \]

2. Simplify using the property \( \ln(e^a) = a \):
   \[
   7x = \ln(27)
   \]

3. Solve for \( x \) by dividing both sides by 7:
   \[
   x = \frac{\ln(27)}{7}
   \]

4. Now, calculate \( \ln(27) \) (using an approximate value of \(\ln(27) \approx 3.2958\)):
   \[
   x \approx \frac{3.2958}{7} \approx 0.471
   \]

Thus, \( x \approx 0.471 \).

Would you like more details, or do you have any questions?

---

### Further Related Questions
1. What is the process for solving other equations of the form \( e^{kx} = C \)?
2. How can logarithmic properties be used to solve exponential equations?
3. Why is the natural logarithm particularly useful for equations involving \( e \)?
4. How would this solution change if the base were not \( e \)?
5. Can you explain the significance of \( e \) in calculus and exponential growth?

### Tip
Remember, the natural logarithm \( \ln(x) \) is the inverse of \( e^x \). This makes it especially useful for solving equations where the exponential function is involved.

Learn how to solve the exponential equation e^{7x} = 27 for x. This problem covers key mathematical concepts, including the use of natural logarithms to isolate variables in exponential equations. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation