solve the equation *\frac{e^{2x\ -3}}{e^{5x\ -7}}\ =\ \frac{15}{10}*

To solve the equation \(\frac{e^{2x-3}}{e^{5x-7}} = \frac{15}{10}\), follow these steps:

1. **Simplify the fraction** on the right-hand side:
   \[
   \frac{15}{10} = \frac{3}{2}
   \]

2. **Rewrite the equation**:
   \[
   \frac{e^{2x-3}}{e^{5x-7}} = \frac{3}{2}
   \]

3. **Combine the exponents** in the fraction on the left-hand side:
   \[
   e^{(2x-3) - (5x-7)} = e^{2x-3 - 5x + 7} = e^{-3x + 4}
   \]
   \[
   e^{-3x + 4} = \frac{3}{2}
   \]

4. **Take the natural logarithm (ln) of both sides** to solve for \(x\):
   \[
   \ln\left(e^{-3x + 4}\right) = \ln\left(\frac{3}{2}\right)
   \]
   \[
   -3x + 4 = \ln\left(\frac{3}{2}\right)
   \]

5. **Solve for \(x\)**:
   \[
   -3x = \ln\left(\frac{3}{2}\right) - 4
   \]
   \[
   x = \frac{4 - \ln\left(\frac{3}{2}\right)}{3}
   \]

Therefore, the solution to the equation \(\frac{e^{2x-3}}{e^{5x-7}} = \frac{3}{2}\) is:
\[
x = \frac{4 - \ln\left(\frac{3}{2}\right)}{3}
\]

Would you like any further details or have any questions?

Here are some questions you might want to ask next:
1. How do you solve logarithmic equations?
2. What are the properties of exponents?
3. Can you explain the natural logarithm function?
4. How do you solve quadratic equations?
5. What is the derivative of an exponential function?

**Tip:** When working with logarithms, remember that \(\ln(e^x) = x\). This property is particularly useful for solving equations involving exponentials.

Learn how to solve the exponential equation (e^(2x-3))/(e^(5x-7)) = 15/10 with a detailed step-by-step solution. This problem covers exponential functions, logarithms, and fraction simplification. Suitable for advanced high school and college students.

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