Solve the equations `log_1000 |x+y| = 1/2 . log_10 y - log_10 |x| = log_100 4` for x and y

Answer

To solve the equations given in the problem, we will break it down step by step.

### Step 1: Solve the first equation
The first equation is:
\[ \log_{1000} |x+y| = \frac{1}{2} \]

We can rewrite this using the change of base formula:
\[ \log_{1000} |x+y| = \frac{\log_{10} |x+y|}{\log_{10} 1000} \]
Since \(1000 = 10^3\), we have:
\[ \log_{10} 1000 = 3 \]
Thus, the equation becomes:
\[ \frac{\log_{10} |x+y|}{3} = \frac{1}{2} \]

Multiplying both sides by 3 gives:
\[ \log_{10} |x+y| = \frac{3}{2} \]

Now, we can exponentiate both sides:
\[ |x+y| = 10^{\frac{3}{2}} = 10 \sqrt{10} \]

### Step 2: Solve the second equation
The second equation is:
\[ \frac{1}{2} \log_{10} y - \log_{10} |x| = \log_{100} 4 \]

We can rewrite \(\log_{100} 4\) as:
\[ \log_{100} 4 = \frac{\log_{10} 4}{\log_{10} 100} \]
Since \(100 = 10^2\), we have:
\[ \log_{10} 100 = 2 \]
Thus, the equation becomes:
\[ \frac{1}{2} \log_{10} y - \log_{10} |x| = \frac{1}{2} \log_{10} 4 \]

Multiplying through by 2 gives:
\[ \log_{10} y - 2 \log_{10} |x| = \log_{10} 4 \]

This can be rewritten as:
\[ \log_{10} \left(\frac{y}{|x|^2}\right) = \log_{10} 4 \]

Exponentiating both sides gives:
\[ \frac{y}{|x|^2} = 4 \]
Thus:
\[ y = 4 |x|^2 \]

### Step 3: Substitute \(y\) into the first equation
Now we substitute \(y = 4 |x|^2\) into the equation \( |x+y| = 10\sqrt{10} \):
\[ |x + 4 |x|^2| = 10\sqrt{10} \]

### Step 4: Consider cases for \(x\)
#### Case 1: \(x \geq 0\)
In this case, \(|x| = x\), so:
\[ |x + 4x^2| = 10\sqrt{10} \]
This simplifies to:
\[ x + 4x^2 = 10\sqrt{10} \]
Rearranging gives:
\[ 4x^2 + x - 10\sqrt{10} = 0 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{-1 \pm \sqrt{1 + 160\sqrt{10}}}{8} \]

#### Case 2: \(x < 0\)
In this case, \(|x| = -x\), so:
\[ |x + 4(-x)^2| = 10\sqrt{10} \]
This simplifies to:
\[ |-x + 4x^2| = 10\sqrt{10} \]
This leads to:
\[ -x + 4x^2 = 10\sqrt{10} \]
Rearranging gives:
\[ 4x^2 - x - 10\sqrt{10} = 0 \]

### Final Step: Solve the quadratic equations
Now we solve both quadratic equations obtained from the cases above to find the values of \(x\) and subsequently \(y\).