The number of real solution(s) of the equation `9^(log_(3)(log_(e )x))=log_(e )x-(log_(e )x)^(2)+1` is equal to

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the equation \( 9^{\log_{3}(\log_{e} x)} = \log_{e} x - (\log_{e} x)^{2} + 1 \), we will follow these steps:

### Step 1: Rewrite the left-hand side
We know that \( 9 = 3^2 \), so we can rewrite the left-hand side:
\[ 9^{\log_{3}(\log_{e} x)} = (3^2)^{\log_{3}(\log_{e} x)} = 3^{2 \cdot \log_{3}(\log_{e} x)} = \log_{e} x \]
Using the property of logarithms, we can simplify this further:
\[ = \log_{e} x^2 \]

### Step 2: Rewrite the right-hand side
The right-hand side is:
\[ \log_{e} x - (\log_{e} x)^{2} + 1 \]
Let \( y = \log_{e} x \). Then we can express the equation as:
\[ \log_{e} x^2 = y - y^2 + 1 \]

### Step 3: Set up the equation
Now substituting \( y \) into the equation gives us:
\[ y^2 = y - y^2 + 1 \]
Rearranging this, we get:
\[ 2y^2 - y + 1 = 0 \]

### Step 4: Solve the quadratic equation
Now we will use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2 \), \( b = -1 \), and \( c = 1 \):
\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} = \frac{1 \pm \sqrt{1 - 8}}{4} = \frac{1 \pm \sqrt{-7}}{4} \]
Since the discriminant \( 1 - 8 = -7 \) is negative, there are no real solutions for \( y \).

### Step 5: Conclusion
Since \( y = \log_{e} x \) has no real solutions, it implies that the original equation has no real solutions.

Thus, the number of real solutions of the equation is:
\[ \boxed{0} \]