Which of the following is not true for equation `x^(2)log8-xlog5=2(log2)-x` (A) equation has one integral root (B) equation has no irrational roots (C) equation has rational roots (D) none of these

Options

A. equation has one integral root
B. equation has no irrational roots
C. equation has rational roots
D. none of these

Explanation

To solve the equation \( x^2 \log 8 - x \log 5 = 2 \log 2 - x \), we will first rearrange it into a standard quadratic form and analyze the roots.

### Step 1: Rearranging the Equation
Start with the given equation:
\[ x^2 \log 8 - x \log 5 = 2 \log 2 - x \]
Rearranging gives:
\[ x^2 \log 8 - x \log 5 + x - 2 \log 2 = 0 \]

### Step 2: Combine Like Terms
We can factor out \( x \) from the terms involving \( x \):
\[ x^2 \log 8 + x(1 - \log 5) - 2 \log 2 = 0 \]

### Step 3: Identify Coefficients
This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where:
- \( a = \log 8 \)
- \( b = 1 - \log 5 \)
- \( c = -2 \log 2 \)

### Step 4: Calculate the Discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (1 - \log 5)^2 - 4(\log 8)(-2 \log 2) \]

### Step 5: Simplify the Discriminant
Calculate \( \log 8 \) and \( \log 2 \):
\[ \log 8 = 3 \log 2 \]
Thus,
\[ D = (1 - \log 5)^2 + 8 (\log 2)^2 \]
Since \( D \) is a sum of squares, \( D \geq 0 \). This means the equation has real roots.

### Step 6: Determine the Nature of Roots
Since the discriminant is non-negative, the roots can be:
- Two distinct real roots if \( D > 0 \)
- One repeated real root if \( D = 0 \)

### Step 7: Check for Integral and Rational Roots
To check if the roots are integral or rational, we can use the Rational Root Theorem. The potential rational roots are factors of \( -2 \log 2 \) divided by factors of \( \log 8 \).

### Step 8: Analyze the Options
- **Option A**: "The equation has one integral root." We need to check if there is an integral root.
- **Option B**: "The equation has no irrational roots." This is likely true since we have determined that the roots are rational.
- **Option C**: "The equation has rational roots." This is likely true as well.
- **Option D**: "None of these." This would be true if all previous options are true.

### Conclusion
After analyzing the roots and their nature, we find that:
- The equation does not have an integral root.
- The equation has rational roots.

Thus, the answer is **(A)**, as it is not true that the equation has one integral root.