https://file.aiquickdraw.com/mathbot/file/1727917166868ebuumoiu.jpg

The equation shown in the image is:

\[
4e^{2x - 3} = 7e^{5 - x}
\]

### Steps to solve:

1. **Divide both sides by \( e^{5-x} \)** to isolate the exponential terms:
   
   \[
   \frac{4e^{2x - 3}}{e^{5-x}} = 7
   \]
   
   Simplifying, this becomes:
   
   \[
   4e^{(2x-3) - (5-x)} = 7
   \]

2. **Simplify the exponent**:

   \[
   (2x-3) - (5-x) = 2x - 3 - 5 + x = 3x - 8
   \]
   
   Substituting back:

   \[
   4e^{3x - 8} = 7
   \]

3. **Divide both sides by 4** to further isolate the exponential term:

   \[
   e^{3x - 8} = \frac{7}{4}
   \]

4. **Take the natural logarithm (ln) of both sides** to remove the exponential:

   \[
   3x - 8 = \ln \left(\frac{7}{4}\right)
   \]

5. **Solve for \( x \)**:

   \[
   3x = \ln \left(\frac{7}{4}\right) + 8
   \]

   \[
   x = \frac{\ln \left(\frac{7}{4}\right) + 8}{3}
   \]

### Final Answer:

\[
x = \frac{\ln \left(\frac{7}{4}\right) + 8}{3}
\]

If you want a numerical solution, we can evaluate this expression:

\[
x \approx \frac{0.5596 + 8}{3} \approx 2.8532
\]

Let me know if you'd like more details or if you have any questions!

### Related Questions:
1. How can we solve other exponential equations using logarithms?
2. What is the general method to isolate the variable in an exponential equation?
3. What are some common properties of natural logarithms used in solving equations?
4. How do changes in the base of exponential functions affect their solutions?
5. How would you graphically represent the solution to this equation?

**Tip:** Always remember to use the properties of logarithms, such as \(\ln(ab) = \ln a + \ln b\), to simplify complex expressions.

Learn how to solve the exponential equation 4e^(2x-3) = 7e^(5-x) using logarithms and properties of exponents. This solution includes step-by-step simplification and application of the natural logarithm. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation