The problem presented is to find the number of real values of $x$ where $x$ is a real number that satisfies the given equation:

$\log x^x = 3 - 2x \log 3$

To solve this, follow these steps:

1. Simplify the equation: $\log x^x = x \log x$ Thus, the equation becomes: $x \log x = 3 - 2x \log 3$

2. Isolate $\log x$: $x \log x + 2x \log 3 = 3$ Factor out $x$ from the left side: $x (\log x + 2 \log 3) = 3$ Simplify further: $x (\log x + \log 9) = 3$ $x \log (9x) = 3$

3. Analyze the function $f(x) = x \log (9x)$: Define $f(x) = x \log (9x)$. We need to find the values of $x$ for which $f(x) = 3$.

4. Find the domain: Since $x \log (9x)$ involves a logarithm, $9x$ must be positive: $9x > 0 \implies x > 0$

5. Behavior of the function:

   • As $x$ approaches $0^+$, $f(x)$ approaches $0$.
   • As $x$ increases, we need to check if $f(x)$ can reach 3.

6. First derivative test: Take the derivative of $f(x)$ to understand its behavior: $f(x) = x \log (9x)$ Use the product rule: $f'(x) = \log (9x) + x \cdot \frac{1}{9x} \cdot 9 = \log (9x) + 1$ $f'(x) = \log (9x) + 1$ For $f'(x) = 0$: $\log (9x) + 1 = 0$ $\log (9x) = -1$ $9x = 10^{-1}$ $x = \frac{1}{9}$

   $f(x)$ is increasing for $x > \frac{1}{9}$ because $f'(x) > 0$.

7. Conclusion: Since $f(x)$ is strictly increasing for $x > \frac{1}{9}$, and $f(x) \to 0$ as $x \to 0^+$, there is exactly one real solution for $x$ such that $f(x) = 3$.

Thus, there is exactly one real value of $x$ that satisfies the given equation.

Do you want more details or have any questions?

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Here are 8 related questions:

1. What is the range of the function $f(x) = x \log (9x)$ for $x > 0$?
2. How does the function $f(x) = x \log (9x)$ behave as $x$ approaches infinity?
3. Can $f(x) = x \log (9x)$ have any complex solutions?
4. How would the equation change if the logarithm had a different base, such as $\log_2$?
5. What happens to the solution if the constant term on the right side is different, for example, $4$ instead of $3$?
6. How can the derivative test be used to find local minima and maxima of $f(x)$?
7. What are the properties of the logarithm function that are useful in solving equations like this one?
8. How can numerical methods be used to approximate solutions to transcendental equations?

Tip: Always remember to check the domain of the logarithmic function to ensure all arguments are positive and valid.