Solution set of the in equality `log_(10^(2)) x-3(log_(10)x)( log_(10)(x-2))+2 log_(10^(2))(x-2) lt 0`, is :

Options

A. `(0,4)`
B. `(-oo,1)`
C. `(4,oo)`
D. `(2, 4)`

Answer

To solve the inequality

\[ \log_{10^2} x - 3(\log_{10} x)(\log_{10} (x-2)) + 2 \log_{10^2} (x-2) < 0, \]

we will follow these steps:

### Step 1: Rewrite the logarithms
First, we can rewrite \(\log_{10^2} x\) and \(\log_{10^2} (x-2)\) using the change of base formula:

\[ \log_{10^2} x = \frac{\log_{10} x}{\log_{10} (10^2)} = \frac{\log_{10} x}{2} \]

and

\[ \log_{10^2} (x-2) = \frac{\log_{10} (x-2)}{2}. \]

Substituting these into the inequality gives us:

\[ \frac{\log_{10} x}{2} - 3(\log_{10} x)(\log_{10} (x-2)) + 2 \cdot \frac{\log_{10} (x-2)}{2} < 0. \]

This simplifies to:

\[ \frac{\log_{10} x}{2} - 3(\log_{10} x)(\log_{10} (x-2)) + \log_{10} (x-2) < 0. \]

### Step 2: Multiply through by 2
To eliminate the fraction, we can multiply the entire inequality by 2 (noting that this does not change the direction of the inequality since 2 is positive):

\[ \log_{10} x - 6(\log_{10} x)(\log_{10} (x-2)) + 2 \log_{10} (x-2) < 0. \]

### Step 3: Rearranging the terms
Rearranging gives us:

\[ \log_{10} x + 2 \log_{10} (x-2) < 6(\log_{10} x)(\log_{10} (x-2)). \]

### Step 4: Let \(y = \log_{10} x\) and \(z = \log_{10} (x-2)\)
Let \(y = \log_{10} x\) and \(z = \log_{10} (x-2)\). Then, we can rewrite the inequality as:

\[ y + 2z < 6yz. \]

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

\[ 6yz - y - 2z > 0. \]

### Step 6: Factor the inequality
Factoring out the common terms, we have:

\[ y(6z - 1) - 2z > 0. \]

### Step 7: Solve for \(y\) and \(z\)
Now we can solve for \(y\) and \(z\):

1. **Case 1:** \(6z - 1 > 0\) or \(z > \frac{1}{6}\) leads to \(y > \frac{2z}{6z - 1}\).
2. **Case 2:** \(6z - 1 < 0\) or \(z < \frac{1}{6}\) leads to \(y < \frac{2z}{6z - 1}\).

### Step 8: Find the solution set
We need to find the values of \(x\) such that \(y = \log_{10} x\) and \(z = \log_{10} (x-2)\) satisfy the above conditions.

### Step 9: Determine the domain
Since \(x\) must be greater than 2 (to ensure \(x-2 > 0\)), we can analyze the behavior of the function in the domain \(x > 2\).

### Step 10: Conclusion
The solution set of the inequality is determined by evaluating the critical points and analyzing the intervals.