For the equation `1 + log_(x) ((4 - x)/(10)) = (log_(10)(log_(10)p) - 1)log_(x)10`, which one of the following(s) is (are) correct?

Options

A. if `p = 10^(3)`, then the equation has two real solutions
B. if `p = 10^(4)`, then the equation has exactly solutions
C. if `p in (10^(4), oo)`, then the equation has no real solutions
D. if `p in (1, 10^(4))`, then the equation has two distinct real solutions.

Explanation

To solve the equation \( 1 + \log_{x} \left( \frac{4 - x}{10} \right) = \left( \log_{10}(\log_{10} p) - 1 \right) \log_{x} 10 \), we will follow these steps:

### Step 1: Rewrite the Equation
We start by rewriting the equation:
\[ 1 + \log_{x} \left( \frac{4 - x}{10} \right) = \left( \log_{10}(\log_{10} p) - 1 \right) \log_{x} 10 \]

### Step 2: Change of Base Formula
Using the change of base formula, we can express the logarithms in terms of base 10:
\[ \log_{x} a = \frac{\log_{10} a}{\log_{10} x} \]
Thus, we rewrite the equation:
\[ 1 + \frac{\log_{10} \left( \frac{4 - x}{10} \right)}{\log_{10} x} = \left( \log_{10}(\log_{10} p) - 1 \right) \frac{\log_{10} 10}{\log_{10} x} \]
Since \(\log_{10} 10 = 1\), we simplify:
\[ 1 + \frac{\log_{10} \left( \frac{4 - x}{10} \right)}{\log_{10} x} = \left( \log_{10}(\log_{10} p) - 1 \right) \frac{1}{\log_{10} x} \]

### Step 3: Multiply Through by \(\log_{10} x\)
To eliminate the denominator, multiply through by \(\log_{10} x\):
\[ \log_{10} x + \log_{10} \left( \frac{4 - x}{10} \right) = \log_{10}(\log_{10} p) - 1 \]

### Step 4: Simplify the Left Side
Using properties of logarithms:
\[ \log_{10} x + \log_{10} (4 - x) - \log_{10} 10 = \log_{10}(\log_{10} p) - 1 \]
This simplifies to:
\[ \log_{10} x + \log_{10} (4 - x) - 1 = \log_{10}(\log_{10} p) - 1 \]
Adding 1 to both sides gives:
\[ \log_{10} x + \log_{10} (4 - x) = \log_{10}(\log_{10} p) \]

### Step 5: Combine Logarithms
Using the property of logarithms:
\[ \log_{10} (x(4 - x)) = \log_{10}(\log_{10} p) \]
This implies:
\[ x(4 - x) = \log_{10} p \]

### Step 6: Rearranging to Form a Quadratic Equation
Rearranging gives:
\[ x^2 - 4x + \log_{10} p = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -4\), and \(c = \log_{10} p\):
\[ x = \frac{4 \pm \sqrt{16 - 4\log_{10} p}}{2} \]
This simplifies to:
\[ x = 2 \pm \sqrt{4 - \log_{10} p} \]

### Step 8: Analyze the Solutions
1. **If \(p = 10^3\)**:
\[ x = 2 \pm \sqrt{4 - 3} = 2 \pm 1 \quad \Rightarrow \quad x = 3 \text{ or } 1 \quad (\text{only } x = 3 \text{ is valid}) \]
2. **If \(p = 10^4\)**:
\[ x = 2 \pm \sqrt{4 - 4} = 2 \quad (\text{only one solution}) \]
3. **If \(p > 10^4\)**:
\[ 4 - \log_{10} p < 0 \quad \Rightarrow \quad \text{no real roots} \]
4. **If \(1 < p < 10^4\)**:
\[ 4 - \log_{10} p > 0 \quad \Rightarrow \quad \text{two distinct solutions} \]

### Conclusion
Based on the analysis:
- \(p = 10^3\) gives one valid solution.
- \(p = 10^4\) gives exactly one solution.
- \(p > 10^4\) gives no real solutions.
- \(1 < p < 10^4\) gives two distinct solutions.