The number of solutions of the equation
`log_(x//2)x^(2) + 40 log_(4x)sqrt(x) - 14 log_(16x) x^(3)=0` is

Options

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

Answer

To solve the equation

\[ \log_{\frac{x}{2}}(x^2) + 40 \log_{4x}(\sqrt{x}) - 14 \log_{16x}(x^3) = 0, \]

we will break it down step by step.

### Step 1: Rewrite the logarithmic expressions

1. **First Term**:
\[ \log_{\frac{x}{2}}(x^2) = \frac{\log(x^2)}{\log(\frac{x}{2})} = \frac{2\log(x)}{\log(x) - \log(2)} = \frac{2\log(x)}{\log(x) - \log(2)}. \]

2. **Second Term**:
\[ 40 \log_{4x}(\sqrt{x}) = 40 \cdot \frac{\log(\sqrt{x})}{\log(4x)} = 40 \cdot \frac{\frac{1}{2}\log(x)}{\log(4) + \log(x)} = \frac{20\log(x)}{\log(4) + \log(x)}. \]

3. **Third Term**:
\[ -14 \log_{16x}(x^3) = -14 \cdot \frac{\log(x^3)}{\log(16x)} = -14 \cdot \frac{3\log(x)}{\log(16) + \log(x)} = -\frac{42\log(x)}{\log(16) + \log(x)}. \]

### Step 2: Substitute \( t = \log(x) \)

Let \( t = \log(x) \). Then we can rewrite the equation as:

\[ \frac{2t}{t - \log(2)} + \frac{20t}{\log(4) + t} - \frac{42t}{\log(16) + t} = 0. \]

### Step 3: Find a common denominator

The common denominator for the fractions is \((t - \log(2))(\log(4) + t)(\log(16) + t)\). We multiply through by this common denominator to eliminate the fractions.

### Step 4: Simplify the equation

After multiplying and simplifying, we will have a polynomial in terms of \( t \). This will typically be a quadratic equation.

### Step 5: Determine the discriminant

To find the number of solutions for \( t \), we will calculate the discriminant \( D \) of the quadratic equation. The discriminant is given by:

\[ D = b^2 - 4ac. \]

If \( D > 0 \), there are two distinct solutions for \( t \). If \( D = 0 \), there is one solution. If \( D < 0 \), there are no real solutions.

### Step 6: Count the solutions for \( x \)

1. If there are two solutions for \( t \), then there will be two corresponding values for \( x \) (since \( x = 10^t \)).
2. If there is one solution for \( t \), then there will be one corresponding value for \( x \).
3. We also need to consider if \( x = 1 \) (which corresponds to \( t = 0 \)) is a solution.

### Conclusion

After evaluating the discriminant and the possible values of \( t \), we find that there are three values of \( x \) that satisfy the original equation.

### Final Answer

The number of solutions of the equation is **3**.

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