To solve the equation \( x^x = 100 \) using logarithms, follow these steps:

1. **Take the natural logarithm of both sides**:
   \[
   \ln(x^x) = \ln(100)
   \]

2. **Simplify the left-hand side** using the property of logarithms \( \ln(a^b) = b \ln(a) \):
   \[
   x \ln(x) = \ln(100)
   \]

3. **Evaluate the right-hand side**:
   \[
   \ln(100) = \ln(10^2) = 2 \ln(10) \approx 2 \times 2.302585 = 4.60517
   \]
   So, the equation simplifies to:
   \[
   x \ln(x) \approx 4.60517
   \]

4. **Solve \( x \ln(x) \approx 4.60517 \)**: This equation does not have a simple algebraic solution, so we use numerical methods to approximate the value of \( x \).

   One way to solve it numerically is to use an iterative method like the Newton-Raphson method. Alternatively, we can use the Lambert W function, which is defined as the inverse function of \( f(W) = W e^W \).

   Rewrite the equation in the form of the Lambert W function:
   \[
   x \ln(x) = 4.60517
   \]
   Let \( y = \ln(x) \), hence \( x = e^y \). Substituting \( x = e^y \) into the equation:
   \[
   y e^y = 4.60517
   \]
   This is of the form \( y e^y = k \), where \( k = 4.60517 \).

5. **Use the Lambert W function**:
   \[
   y = W(4.60517)
   \]

6. **Evaluate the Lambert W function**:
   The Lambert W function \( W(z) \) for \( z = 4.60517 \) can be evaluated numerically. Using a numerical calculator or software that can compute the Lambert W function, we find:
   \[
   W(4.60517) \approx 1.385
   \]

7. **Solve for \( x \)**:
   Recall \( y = \ln(x) \):
   \[
   \ln(x) \approx 1.385
   \]
   Thus,
   \[
   x \approx e^{1.385} \approx 3.994
   \]

Therefore, the approximate solution to \( x^x = 100 \) is \( x \approx 3.994 \).

Would you like more details or have any questions about this process?

Here are 8 relative questions that expand on this topic:
1. What are the properties of the natural logarithm that are useful in solving equations?
2. How does the Newton-Raphson method work for solving nonlinear equations?
3. What is the Lambert W function and how is it used in mathematics?
4. Can the equation \( x^x = k \) be solved for any value of \( k \)?
5. How do you evaluate the Lambert W function without a calculator?
6. What are other numerical methods for solving transcendental equations?
7. How accurate is the solution \( x \approx 3.994 \) for \( x^x = 100 \)?
8. What are some practical applications of solving equations like \( x^x = 100 \)?

**Tip:** When solving equations involving exponents and logarithms, always consider properties and transformations that simplify the equation, such as using logarithmic identities and numerical methods for approximation.

Learn how to solve the equation x^x = 100 using logarithms and the Lambert W function. This problem involves advanced mathematical concepts including exponential functions, logarithms, and numerical methods for solving transcendental equations.

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