how to determine all x-values that satisfy the following equation: 4x-3=e^2ln(2-x)

To determine the \(x\)-values that satisfy the equation \( 4x - 3 = e^{2\ln(2 - x)} \), follow these steps:

### Step 1: Simplify the equation
Start with the given equation:
\[
4x - 3 = e^{2\ln(2 - x)}
\]

We can simplify the right-hand side using the property of logarithms: \( e^{a\ln(b)} = b^a \). Thus:
\[
e^{2\ln(2 - x)} = (2 - x)^2
\]

The equation now becomes:
\[
4x - 3 = (2 - x)^2
\]

### Step 2: Expand and rearrange the equation
Expand the right-hand side:
\[
(2 - x)^2 = 4 - 4x + x^2
\]

Now, rewrite the equation as:
\[
4x - 3 = 4 - 4x + x^2
\]

### Step 3: Move all terms to one side
Bring all terms to one side of the equation:
\[
x^2 - 8x + 7 = 0
\]

### Step 4: Solve the quadratic equation
This is now a quadratic equation in the form \(x^2 - 8x + 7 = 0\). We can solve it using the quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(7)}}{2(1)}
\]
Simplifying:
\[
x = \frac{8 \pm \sqrt{64 - 28}}{2} = \frac{8 \pm \sqrt{36}}{2}
\]
\[
x = \frac{8 \pm 6}{2}
\]

### Step 5: Find the solutions
Now, calculate the two possible values for \(x\):
\[
x = \frac{8 + 6}{2} = 7 \quad \text{or} \quad x = \frac{8 - 6}{2} = 1
\]

### Step 6: Check for any restrictions
Since the original equation involves \( \ln(2 - x) \), we must ensure that \( 2 - x > 0 \) (because the logarithm is only defined for positive arguments). This gives:
\[
x < 2
\]

Thus, the only valid solution is \( x = 1 \).

### Final Answer:
The solution to the equation is:
\[
x = 1
\]

---

Would you like more details on any step? Here are five related questions to expand on this:

1. How does the property \(e^{a \ln(b)} = b^a\) work?
2. Can you show more examples of solving quadratic equations?
3. How do you solve logarithmic equations involving multiple logs?
4. What are the restrictions when solving equations with logarithms?
5. How do you interpret complex roots in quadratic equations?

**Tip:** Always check for restrictions, especially when dealing with logarithmic or square root functions, as their domains may limit the possible solutions.

How to determine all x-values that satisfy the following equation: 4x - 3 = e^2ln(2 - x)

Learn how to solve the equation 4x - 3 = e^2ln(2 - x) by simplifying logarithmic expressions and using the quadratic formula. This step-by-step solution highlights exponential and logarithmic properties with domain restrictions, suitable for Grades 10-12 students.

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