The number of real solutions of equation `x^(log_(x)2)+x^(2)=3x` is

Explanation

To solve the equation \( x^{\log_{x}2} + x^2 = 3x \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^{\log_{x}2} + x^2 = 3x \]
Using the property of logarithms, we can rewrite \( \log_{x}2 \) as \( \frac{\log_{10}2}{\log_{10}x} \). However, for simplicity, we can also express \( x^{\log_{x}2} \) directly in terms of base 2:
\[ x^{\log_{x}2} = 2 \]
This is because \( x^{\log_{x}2} \) simplifies to \( 2 \) when we consider the properties of logarithms.

### Step 2: Substitute and simplify
Now substituting \( x^{\log_{x}2} \) with \( 2 \):
\[ 2 + x^2 = 3x \]
Rearranging the equation gives:
\[ x^2 - 3x + 2 = 0 \]

### Step 3: Factor the quadratic equation
Next, we factor the quadratic equation:
\[ (x - 1)(x - 2) = 0 \]
This gives us the potential solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 4: Analyze the solutions
We have two potential solutions: \( x = 1 \) and \( x = 2 \). However, we need to check the validity of these solutions in the context of the original equation.

Since \( x \) is in the base of the logarithm, \( x \) cannot equal \( 1 \) (as \( \log_{1}2 \) is undefined). Therefore, we discard \( x = 1 \).

### Step 5: Conclusion
The only valid solution is:
\[ x = 2 \]
Thus, the number of real solutions to the equation \( x^{\log_{x}2} + x^2 = 3x \) is:
\[ \boxed{1} \]

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