The equation `x^3/4((log)_2x)^(2+(log)_2x-5/4)=sqrt(2)` has at least one real solution exactly three solutions exactly one irrational solution complex roots

Options

A. at least one real solution
B. exactly three solutions
C. exactly one irrational solution
D. complex roots

Answer

To solve the equation

\[ \frac{x^{3}}{4} \left( \log_2 x \right)^{2 + \log_2 x - \frac{5}{4}} = \sqrt{2} \]

we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:

\[ \frac{x^{3}}{4} \left( \log_2 x \right)^{2 + \log_2 x - \frac{5}{4}} = \sqrt{2} \]

### Step 2: Take logarithm on both sides
We take logarithm (base 2) of both sides. Using the property of logarithms, we can express the left-hand side as:

\[ \log_2 \left( \frac{x^{3}}{4} \left( \log_2 x \right)^{2 + \log_2 x - \frac{5}{4}} \right) = \log_2 \left( \sqrt{2} \right) \]

### Step 3: Simplify the logarithm
Using the properties of logarithms, we can simplify the left-hand side:

\[ \log_2 \left( \frac{x^{3}}{4} \right) + \log_2 \left( \left( \log_2 x \right)^{2 + \log_2 x - \frac{5}{4}} \right) = \frac{1}{2} \]

This gives us:

\[ \log_2 x^{3} - \log_2 4 + \left( 2 + \log_2 x - \frac{5}{4} \right) \log_2 \left( \log_2 x \right) = \frac{1}{2} \]

### Step 4: Let \( t = \log_2 x \)
Let \( t = \log_2 x \), then we can rewrite the equation:

\[ 3t - 2 + \left( 2 + t - \frac{5}{4} \right) \log_2 t = \frac{1}{2} \]

### Step 5: Rearranging the equation
Rearranging gives us:

\[ 3t + \left( t - \frac{1}{4} \right) \log_2 t = \frac{5}{2} \]

### Step 6: Solve for \( t \)
This is a transcendental equation in \( t \). We can analyze it to find possible values for \( t \).

### Step 7: Finding roots
We can use numerical methods or graphical methods to find the roots of this equation.

### Step 8: Back substitute to find \( x \)
Once we have the values of \( t \), we can find \( x \) using:

\[ x = 2^t \]

### Step 9: Analyze the nature of solutions
We will analyze the number of solutions based on the behavior of the function derived from our equation.

### Conclusion
After performing the above steps, we can conclude that the equation has:

- **At least one real solution** (since we found at least one root).
- **Exactly three solutions** (if the cubic equation derived has three distinct real roots).
- **Exactly one irrational solution** (if one of the roots is irrational).
- **Complex roots** (if the discriminant of the cubic equation is negative).