Solution(s) of the equation. `3x^2-2x^3 = log_2 (x^2 + 1) - log_2 x` is/are

Options

A. 1
B. 2
C. 3
D. None of these

Answer

To solve the equation \(3x^2 - 2x^3 = \log_2(x^2 + 1) - \log_2 x\), we will follow these steps:

### Step 1: Simplify the logarithmic expression
Using the property of logarithms, we can combine the two logarithmic terms on the right side:
\[ \log_2(x^2 + 1) - \log_2 x = \log_2\left(\frac{x^2 + 1}{x}\right) \]
This simplifies to:
\[ \log_2\left(x + \frac{1}{x}\right) \]

### Step 2: Rewrite the equation
Now we can rewrite the original equation as:
\[ 3x^2 - 2x^3 = \log_2\left(x + \frac{1}{x}\right) \]

### Step 3: Convert to exponential form
To eliminate the logarithm, we can exponentiate both sides:
\[ 2^{3x^2 - 2x^3} = x + \frac{1}{x} \]

### Step 4: Test a potential solution
Let's test \(x = 1\):
\[ 3(1)^2 - 2(1)^3 = 3 - 2 = 1 \]
And for the right side:
\[ \log_2(1 + 1) = \log_2(2) = 1 \]
Both sides are equal, so \(x = 1\) is indeed a solution.

### Step 5: Verify if there are other solutions
To check if there are any other solutions, we can analyze the behavior of both sides of the equation. The left side \(3x^2 - 2x^3\) is a cubic polynomial, and the right side \(x + \frac{1}{x}\) is a function that behaves differently.

### Step 6: Analyze the function
The left side \(3x^2 - 2x^3\) is a downward-opening cubic function, while the right side \(x + \frac{1}{x}\) is always greater than or equal to 2 for \(x > 0\). Since we have already found that \(x = 1\) is a solution, we can conclude that it is the only solution in the positive domain.

### Final Solution
Thus, the solution to the equation \(3x^2 - 2x^3 = \log_2(x^2 + 1) - \log_2 x\) is:
\[ \boxed{1} \]