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To solve this problem using Newton's method, we start by defining the function \( f(x) \) from the given equation:

### Step 1: Define the Function
The equation given is:
\[
\ln(45x) = 33x^3 - 12e
\]

Rearrange this equation to bring all terms to one side:
\[
f(x) = \ln(45x) - 33x^3 + 12e
\]

### Step 2: Compute the Derivative
We need the derivative of \( f(x) \), which is:
\[
f'(x) = \frac{d}{dx} \left( \ln(45x) - 33x^3 + 12e \right)
\]

Using the chain rule and the derivative of each term:
\[
f'(x) = \frac{1}{45x} \cdot 45 - 99x^2 = \frac{1}{x} - 99x^2
\]

### Step 3: Newton's Method Formula
Newton's method formula for the next approximation \( x_{n+1} \) is:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

### Step 4: Initial Estimate
Given \( x_0 = 1 \), we will compute three iterations to find \( x_3 \).

Now, let's calculate each iteration step-by-step:

1. **Iteration 1: Compute \( x_1 \)**
\[
f(x_0) = \ln(45 \cdot 1) - 33 \cdot 1^3 + 12e
\]
\[
f'(x_0) = \frac{1}{1} - 99 \cdot 1^2
\]
\[
x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}
\]

2. **Iteration 2: Compute \( x_2 \)**
\[
f(x_1) = \ln(45x_1) - 33x_1^3 + 12e
\]
\[
f'(x_1) = \frac{1}{x_1} - 99x_1^2
\]
\[
x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}
\]

3. **Iteration 3: Compute \( x_3 \)**
\[
f(x_2) = \ln(45x_2) - 33x_2^3 + 12e
\]
\[
f'(x_2) = \frac{1}{x_2} - 99x_2^2
\]
\[
x_3 = x_2 - \frac{f(x_2)}{f'(x_2)}
\]

### Step 5: Calculate Values
I will now compute these values.After applying three iterations of Newton's method, the approximate solution \( x_3 \) is:

\[
x_3 \approx 1.0338
\]

### Detailed Iteration Values:
1. \( x_0 = 1 \) (initial estimate)
2. \( x_1 \approx 1.0349596371 \)
3. \( x_2 \approx 1.0337890393 \)
4. \( x_3 \approx 1.0337876945 \)

The answer is accurate to four decimal places, as required.

Would you like further details or explanations? Here are some questions that might help you explore more:

1. What are the underlying principles of Newton's method for finding roots?
2. How do the convergence properties of Newton's method depend on the choice of the initial estimate?
3. Can Newton's method fail to converge? If so, under what conditions?
4. How does the function's behavior affect the convergence speed of Newton's method?
5. What are some real-world applications of Newton's method?

**Tip:** Always ensure your initial estimate is close to the expected root to improve the convergence rate of Newton's method.

Apply three iterations of Newton's method to find an approximate solution of the equation ln(45x) = 33x^3 - 12e. Use an initial estimate of x_0 = 1.

Learn how to apply Newton's method to find the approximate solution of the equation ln(45x) = 33x^3 - 12e. The problem involves iterative calculation using Newton's formula, and the solution is accurate to four decimal places after three iterations.

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