which of the statements are accurate in solving for x: e^(3-2x) = 4
1. x = 3/2 - (in 4)/2
2. in 4 is a constant
3. x = 3/2 - in 2
4. taking log on both sides is a good first step
5. 3 - in 4

Let's analyze each of the statements for solving the equation \( e^{3-2x} = 4 \):

1. **\( x = \frac{3}{2} - \frac{\ln 4}{2} \)**:
   - This statement is accurate. To solve the equation, you would take the natural logarithm (ln) on both sides, leading to \( 3 - 2x = \ln 4 \). Then, solve for \( x \) by isolating it: \( 2x = 3 - \ln 4 \), so \( x = \frac{3}{2} - \frac{\ln 4}{2} \).

2. **\( \ln 4 \) is a constant**:
   - This statement is accurate. \( \ln 4 \) is a constant since it is the natural logarithm of 4, which evaluates to a specific numerical value.

3. **\( x = \frac{3}{2} - \ln 2 \)**:
   - This statement is not accurate. The correct expression for \( x \) involves \( \ln 4 \), not \( \ln 2 \). \( x = \frac{3}{2} - \frac{\ln 4}{2} \) is correct, but \( \ln 4 \) is not the same as \( \ln 2 \).

4. **Taking log on both sides is a good first step**:
   - This statement is accurate. Taking the natural logarithm on both sides helps to simplify the equation by getting rid of the exponential function, making it easier to solve for \( x \).

5. **\( 3 - \ln 4 \)**:
   - This statement is partially accurate, but it’s not a complete equation. \( 3 - \ln 4 \) is part of the expression for \( x \), but without context, it doesn't directly contribute to solving for \( x \). It's the result of simplifying \( 3 - 2x = \ln 4 \) before dividing by 2.

**Accurate statements**: 1, 2, 4.

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Would you like more details or have any questions? Here are some related questions:

1. How do you solve logarithmic equations involving exponents?
2. What is the process for taking the natural logarithm on both sides of an equation?
3. Why is \( \ln 4 \) considered a constant, and what is its approximate value?
4. How does the natural logarithm help in solving exponential equations?
5. What is the relationship between \( \ln 4 \) and \( \ln 2 \)?
6. Can \( x = \frac{3}{2} - \frac{\ln 4}{2} \) be simplified further?
7. What are some common mistakes to avoid when solving logarithmic equations?
8. How do you check the accuracy of your solution after solving for \( x \)?

**Tip**: When dealing with exponential equations, always consider taking the natural logarithm to simplify the equation, as it helps to linearize the exponent.

Explore how to solve the exponential equation e^(3-2x) = 4 step-by-step. Understand the use of logarithms in simplifying the equation and find the accurate solution for x. Suitable for high school students learning about exponential functions and logarithms.

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