The equation `x^([(log_(3)x)^(2)-(9//2)log_(3)x+5])=3sqrt(3)` has

Options

A. at least one real solution
B. exactly three real solutions
C. exactly one irrational solution
D. none of these

Answer

To solve the equation \( x^{\left( (\log_3 x)^2 - \frac{9}{2} \log_3 x + 5 \right)} = 3\sqrt{3} \), we will follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the logarithm base 3 of both sides to simplify the equation.

\[ \log_3 \left( x^{\left( (\log_3 x)^2 - \frac{9}{2} \log_3 x + 5 \right)} \right) = \log_3 (3\sqrt{3}) \]

### Step 2: Apply the logarithmic property
Using the property of logarithms that states \( \log_b (a^c) = c \cdot \log_b a \), we can rewrite the left side:

\[ \left( (\log_3 x)^2 - \frac{9}{2} \log_3 x + 5 \right) \cdot \log_3 x = \log_3 (3^{3/2}) \]

### Step 3: Simplify the right side
The right side simplifies as follows:

\[ \log_3 (3^{3/2}) = \frac{3}{2} \]

### Step 4: Set up the equation
Now we have:

\[ \left( (\log_3 x)^2 - \frac{9}{2} \log_3 x + 5 \right) \cdot \log_3 x = \frac{3}{2} \]

Let \( t = \log_3 x \). Substituting \( t \) into the equation gives:

\[ (t^2 - \frac{9}{2} t + 5) t = \frac{3}{2} \]

### Step 5: Rearranging the equation
This expands to:

\[ t^3 - \frac{9}{2} t^2 + 5t - \frac{3}{2} = 0 \]

### Step 6: Clear the fraction
To eliminate the fraction, multiply the entire equation by 2:

\[ 2t^3 - 9t^2 + 10t - 3 = 0 \]

### Step 7: Factor the polynomial
We can factor this polynomial. We can try to find rational roots using the Rational Root Theorem or synthetic division. Testing possible roots, we find:

\[ (t - 1)(2t^2 - 7t + 3) = 0 \]

### Step 8: Solve the quadratic equation
Now we solve the quadratic \( 2t^2 - 7t + 3 = 0 \) using the quadratic formula:

\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2} \]

Calculating the discriminant:

\[ b^2 - 4ac = 49 - 24 = 25 \]

Thus,

\[ t = \frac{7 \pm 5}{4} \]

This gives us two solutions:

1. \( t = \frac{12}{4} = 3 \)
2. \( t = \frac{2}{4} = \frac{1}{2} \)

### Step 9: Find \( x \)
Recall that \( t = \log_3 x \). Therefore, we have:

1. \( \log_3 x = 1 \) implies \( x = 3^1 = 3 \)
2. \( \log_3 x = 3 \) implies \( x = 3^3 = 27 \)
3. \( \log_3 x = \frac{1}{2} \) implies \( x = 3^{1/2} = \sqrt{3} \)

### Step 10: Conclusion
Thus, the solutions to the equation are:

- \( x = 3 \)
- \( x = 27 \)
- \( x = \sqrt{3} \)