Let W,X,Y and Z be positive real number such that
`log(W.Z)+log(W.Y)=, log(Y.Z)+log(Y.X)+log(X.Z)=4`
`log(W.Z) + log(W.Y)=2, log(Y.Z)+log(Y.X)=3, log(X.Z)=4`.
The value of the product (WXYZ) equals (base of the log is 10)

Options

A. `10^(2)`
B. `10^(3)`
C. `10^(4)`
D. `10^(9)`

Answer

To solve the problem, we will use the properties of logarithms to derive the value of the product \( WXYZ \).

### Given:
1. \( \log(WZ) + \log(WY) = 2 \)
2. \( \log(YZ) + \log(YX) = 3 \)
3. \( \log(XW) + \log(XZ) = 4 \)

### Step 1: Rewrite the logarithmic equations
Using the property of logarithms that states \( \log(A) + \log(B) = \log(AB) \), we can rewrite the given equations:

1. From \( \log(WZ) + \log(WY) = 2 \):
\[ \log(W^2ZY) = 2 \implies W^2ZY = 10^2 = 100 \quad \text{(Equation 1)} \]

2. From \( \log(YZ) + \log(YX) = 3 \):
\[ \log(Y^2ZX) = 3 \implies Y^2ZX = 10^3 = 1000 \quad \text{(Equation 2)} \]

3. From \( \log(XW) + \log(XZ) = 4 \):
\[ \log(X^2WZ) = 4 \implies X^2WZ = 10^4 = 10000 \quad \text{(Equation 3)} \]

### Step 2: Multiply the equations
Now we will multiply the three equations obtained:

\[ (W^2ZY)(Y^2ZX)(X^2WZ) = 100 \times 1000 \times 10000 \]

Calculating the right-hand side:
\[ 100 \times 1000 = 100000 \quad \text{and} \quad 100000 \times 10000 = 1000000000 = 10^9 \]

### Step 3: Simplify the left-hand side
Now simplifying the left-hand side:
\[ W^2ZY \cdot Y^2ZX \cdot X^2WZ = W^3Z^3Y^3X \]

### Step 4: Set up the equation
We can now set up the equation:
\[ W^3Z^3Y^3X = 10^9 \]

### Step 5: Solve for \( WXYZ \)
Taking the cube root of both sides:
\[ (WXYZ)^3 = 10^9 \implies WXYZ = 10^{9/3} = 10^3 = 1000 \]

### Conclusion
Thus, the value of the product \( WXYZ \) is \( 1000 \).

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