To solve the given logarithmic equation step by step, we start with the equation:
\[
\log_{10}|x^3 + y^3| - \log_{10}|x^2 - xy + y^2| + \log_{10}|x^3 - y^3| - \log_{10}|x^2 + xy + y^2| = \log_{10}221
\]
### Step 1: Combine the logarithmic expressions using properties of logarithms
Using the properties of logarithms, we can combine the terms:
\[
\log_{10} \left( \frac{|x^3 + y^3| \cdot |x^3 - y^3|}{|x^2 - xy + y^2| \cdot |x^2 + xy + y^2|} \right) = \log_{10} 221
\]
### Step 2: Remove the logarithm by exponentiating both sides
Since the logarithm is equal, we can exponentiate both sides to eliminate the logarithm:
\[
\frac{|x^3 + y^3| \cdot |x^3 - y^3|}{|x^2 - xy + y^2| \cdot |x^2 + xy + y^2|} = 221
\]
### Step 3: Use the formulas for sums and differences of cubes
Recall the formulas:
- \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \)
- \( x^3 - y^3 = (x - y)(x^2 + xy + y^2) \)
Substituting these into our equation gives:
\[
\frac{(x + y)(x^2 - xy + y^2) \cdot (x - y)(x^2 + xy + y^2)}{|x^2 - xy + y^2| \cdot |x^2 + xy + y^2|} = 221
\]
### Step 4: Simplify the equation
The terms \( |x^2 - xy + y^2| \) and \( |x^2 + xy + y^2| \) cancel out, leading to:
\[
|(x + y)(x - y)| = 221
\]
### Step 5: Solve for \( x \) and \( y \)
This implies two cases:
1. \( (x + y)(x - y) = 221 \)
2. \( (x + y)(x - y) = -221 \)
### Step 6: Analyze the cases
#### Case 1: \( (x + y)(x - y) = 221 \)
Let \( a = x + y \) and \( b = x - y \). Then:
\[
ab = 221
\]
The integer factor pairs of 221 are \( (1, 221), (13, 17), (-1, -221), (-13, -17) \).
From these pairs, we can find \( x \) and \( y \):
1. For \( (1, 221) \):
- \( x + y = 221 \)
- \( x - y = 1 \)
- Solving gives \( x = 111, y = 110 \).
2. For \( (13, 17) \):
- \( x + y = 17 \)
- \( x - y = 13 \)
- Solving gives \( x = 15, y = 2 \).
3. For \( (-1, -221) \):
- \( x + y = -221 \)
- \( x - y = -1 \)
- Solving gives \( x = -111, y = -110 \).
4. For \( (-13, -17) \):
- \( x + y = -17 \)
- \( x - y = -13 \)
- Solving gives \( x = -15, y = -2 \).
#### Case 2: \( (x + y)(x - y) = -221 \)
Similar factor pairs lead to:
1. For \( (1, -221) \):
- \( x + y = -221 \)
- \( x - y = 1 \)
- Solving gives \( x = -110, y = -111 \).
2. For \( (13, -17) \):
- \( x + y = -17 \)
- \( x - y = 13 \)
- Solving gives \( x = -2, y = -15 \).
### Step 7: Finding specific values
(i) If \( x = 111 \):
- From the first case, \( y \) can be \( 110 \) or \( -110 \).
(ii) If \( y = 2 \):
- From the second case, \( x \) can be \( 15 \) or \( -15 \).
### Final Answers:
- (i) If \( x = 111 \), then \( y \) can be \( 110 \) or \( -110 \).
- (ii) If \( y = 2 \), then \( x \) can be \( 15 \) or \( -15 \).
