Let G,O,E and L be positive real numbers such that log(G.L)+log(G.E)=3,log(E.L)+log(E.O)=4, log(O.G)+log(O.L)=5 (base of the log is 10)
If `log(G/O)` and `log(O /E)` are the roots of the equation

Options

A. `x^2+x=0`
B. `x^2-x=0`
C. `x^2-2x+3=0`
D. `x^2-1=0`

Answer

To solve the problem step by step, we start with the given equations involving logarithms.

### Step 1: Rewrite the given logarithmic equations
We have the following equations:
1. \( \log(G \cdot L) + \log(G \cdot E) = 3 \)
2. \( \log(E \cdot L) + \log(E \cdot O) = 4 \)
3. \( \log(O \cdot G) + \log(O \cdot L) = 5 \)

Using the property of logarithms that states \( \log(a) + \log(b) = \log(a \cdot b) \), we can rewrite these equations as:
1. \( \log(G^2 \cdot L \cdot E) = 3 \)
2. \( \log(E^2 \cdot L \cdot O) = 4 \)
3. \( \log(O^2 \cdot G \cdot L) = 5 \)

### Step 2: Convert logarithmic equations to exponential form
From the logarithmic equations, we can convert them to exponential form:
1. \( G^2 \cdot L \cdot E = 10^3 = 1000 \) (Equation 1)
2. \( E^2 \cdot L \cdot O = 10^4 = 10000 \) (Equation 2)
3. \( O^2 \cdot G \cdot L = 10^5 = 100000 \) (Equation 3)

### Step 3: Divide the equations to find ratios
To find the ratios \( \frac{G}{O} \) and \( \frac{O}{E} \), we will divide the equations.

**Dividing Equation 1 by Equation 3:**
\[ \frac{G^2 \cdot L \cdot E}{O^2 \cdot G \cdot L} = \frac{1000}{100000} \]
This simplifies to:
\[ \frac{G}{O^2} \cdot E = \frac{1}{100} \]
Thus, we have:
\[ \frac{G \cdot E}{O^2} = \frac{1}{100} \quad \text{(Equation 4)} \]

**Dividing Equation 1 by Equation 2:**
\[ \frac{G^2 \cdot L \cdot E}{E^2 \cdot L \cdot O} = \frac{1000}{10000} \]
This simplifies to:
\[ \frac{G^2}{E^2} \cdot \frac{1}{O} = \frac{1}{10} \]
Thus, we have:
\[ \frac{G^2}{E^2} = \frac{O}{10} \quad \text{(Equation 5)} \]

### Step 4: Multiply the equations to find \( G \) and \( O \)
Now, we multiply Equation 4 and Equation 5:
\[ \frac{G \cdot E}{O^2} \cdot \frac{G^2}{E^2} = \frac{1}{100} \cdot \frac{O}{10} \]
This simplifies to:
\[ \frac{G^3}{O^2 \cdot E} = \frac{1}{1000} \]
From this, we can express \( \frac{G}{O} \):
\[ \frac{G^3}{O^3} = \frac{1}{1000} \implies \frac{G}{O} = \frac{1}{10} \]

### Step 5: Find \( \log\left(\frac{G}{O}\right) \)
Taking the logarithm:
\[ \log\left(\frac{G}{O}\right) = \log\left(\frac{1}{10}\right) = -1 \quad \text{(Equation 6)} \]

### Step 6: Find \( \frac{O}{E} \)
Next, we find \( \frac{O}{E} \) using Equation 2 and Equation 3:
**Dividing Equation 2 by Equation 3:**
\[ \frac{E^2 \cdot L \cdot O}{O^2 \cdot G \cdot L} = \frac{10000}{100000} \]
This simplifies to:
\[ \frac{E^2}{O^2} \cdot \frac{1}{G} = \frac{1}{10} \]
Thus, we have:
\[ \frac{E^2}{O^2} = \frac{G}{10} \quad \text{(Equation 7)} \]

### Step 7: Multiply Equation 4 and Equation 7
We can find \( \frac{O^3}{E^3} \) similarly:
\[ \frac{G \cdot E}{O^2} \cdot \frac{E^2}{O^2} = \frac{1}{100} \cdot \frac{G}{10} \]
This simplifies to:
\[ \frac{E^3}{O^3} = \frac{1}{1000} \]
Thus, we find:
\[ \frac{O}{E} = 10 \]

### Step 8: Find \( \log\left(\frac{O}{E}\right) \)
Taking the logarithm:
\[ \log\left(\frac{O}{E}\right) = \log(10) = 1 \quad \text{(Equation 8)} \]

### Step 9: Form the quadratic equation
The roots of the equation are \( -1 \) and \( 1 \). Therefore, the quadratic equation can be written as:
\[ x^2 - 1 = 0 \implies x^2 - 1 = 0 \]

### Final Answer
The equation whose roots are \( \log\left(\frac{G}{O}\right) \) and \( \log\left(\frac{O}{E}\right) \) is:
\[ x^2 - 1 = 0 \]