To solve the exponential equation \( e^{1 - 2x} = e^{3x - 5} \), follow these steps:
### Step 1: Recognize the Equality of Exponents
Since the bases of the exponentials are the same (both are \( e \)), we can equate their exponents directly. So, we have:
[tex]\[
1 - 2x = 3x - 5
\][/tex]
### Step 2: Solve the Linear Equation
To solve the equation \( 1 - 2x = 3x - 5 \), first combine like terms:
1. Move all terms involving \( x \) to one side by adding \( 2x \) to both sides:
[tex]\[
1 = 3x - 5 + 2x
\][/tex]
[tex]\[
1 = 5x - 5
\][/tex]
2. Move the constant term to the other side by adding 5 to both sides:
[tex]\[
1 + 5 = 5x
\][/tex]
[tex]\[
6 = 5x
\][/tex]
3. Solve for \( x \) by dividing both sides by 5:
[tex]\[
x = \frac{6}{5}
\][/tex]
### Step 3: Identify All Solutions
The equation \( e^{1 - 2x} = e^{3x - 5} \) might also have complex solutions due to the periodic nature of exponential functions in the complex plane. These solutions are:
[tex]\[ x = \frac{6}{5}, \quad \log\left(-\frac{\sqrt{5} e^{\frac{6}{5}}}{4} - \frac{e^{\frac{6}{5}}}{4} - i\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} e^{\frac{6}{5}}\right), \quad \log\left(-\frac{\sqrt{5} e^{\frac{6}{5}}}{4} - \frac{e^{\frac{6}{5}}}{4} + i\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} e^{\frac{6}{5}}\right), \][/tex]
[tex]\[ \log\left(-\frac{e^{\frac{6}{5}}}{4} + \frac{\sqrt{5} e^{\frac{6}{5}}}{4} - i\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} e^{\frac{6}{5}}\right), \quad \log\left(-\frac{e^{\frac{6}{5}}}{4} + \frac{\sqrt{5} e^{\frac{6}{5}}}{4} + i\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} e^{\frac{6}{5}}\right) \][/tex]
### Final Solution
The solutions to the equation \( e^{1 - 2x} = e^{3x - 5} \) are:
1. \( x = \frac{6}{5} \)
2. \( x = \log\left(-\frac{\sqrt{5} e^{\frac{6}{5}}}{4} - \frac{e^{\frac{6}{5}}}{4} - i\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} e^{\frac{6}{5}}\right) \)
3. \( x = \log\left(-\frac{\sqrt{5} e^{\frac{6}{5}}}{4} - \frac{e^{\frac{6}{5}}}{4} + i\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} e^{\frac{6}{5}}\right) \)
4. \( x = \log\left(-\frac{e^{\frac{6}{5}}}{4} + \frac{\sqrt{5} e^{\frac{6}{5}}}{4} - i\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} e^{\frac{6}{5}}\right) \)
5. \( x = \log\left(-\frac{e^{\frac{6}{5}}}{4} + \frac{\sqrt{5} e^{\frac{6}{5}}}{4} + i\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} e^{\frac{6}{5}}\right) \)
Thus, these represent the complete set of solutions.
