What is the value of x raise to power log x

The expression \( x^{\log x} \) can be simplified using logarithmic properties. 

Here’s how:

1. First, recall the logarithmic identity:
   \[
   a^{\log_b(x)} = x^{\log_b(a)}
   \]
   However, for this case, we do not have a base mentioned, so we assume the logarithm is base \( 10 \) or natural log \( e \).

2. Let's express the natural logarithm version \( x^{\log x} \). Rewrite \( \log x \) as the natural logarithm \( \ln x \) (logarithm base \( e \)):

   \[
   x^{\ln x}
   \]

3. Taking the natural logarithm of both sides, we get:

   \[
   \ln(x^{\ln x}) = \ln x \cdot \ln x = (\ln x)^2
   \]

Thus, \( x^{\log x} \) simplifies to \( e^{(\ln x)^2} \), showing the value depends on the form of the logarithm used.

Do you have any clarifications or need more details?

### Follow-up Questions:
1. What is the result if \( x = e \) in the expression \( x^{\log x} \)?
2. How does the value change if the base of the logarithm is 10 instead of \( e \)?
3. How can the change of base formula be applied to this expression?
4. What happens when \( x \) is a negative value or a fraction in \( x^{\log x} \)?
5. What is the graph of \( y = x^{\log x} \) for different values of \( x \)?

**Tip:** For logarithmic and exponential functions, understanding base changes can simplify complex expressions.

What is the value of x raised to power log x?

Explore how to simplify the expression x raised to the power of log(x) using logarithmic identities and natural logarithms. This solution covers the key concepts of exponents, natural logarithms, and the change of base formula, making it ideal for advanced high school or early college students.

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