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 the minimum value of 3G+2L+2O+E is `2^(lamda)3^mu5^nu` ,Where `lamda,mu`,and `nu` are whole numbers, the value of `sum(lamda^(mu)+mu^(nu))` is

Options

A. 7
B. 13
C. 19
D. None of these

Answer

To solve the problem step by step, we start with the given logarithmic equations and simplify them to find the minimum value of the expression \(3G + 2L + 2O + E\).

### Step 1: Write down the equations
We are given three equations based on logarithmic properties:
1. \( \log(G \cdot L) + \log(G \cdot E) = 3 \)
2. \( \log(E \cdot L) + \log(E \cdot O) = 4 \)
3. \( \log(O \cdot G) + \log(O \cdot L) = 5 \)

### Step 2: Simplify using logarithmic properties
Using the property that \(\log(a) + \log(b) = \log(a \cdot b)\), we can rewrite the equations:
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 3: Convert logarithmic equations to exponential form
From the logarithmic equations, we can express them in 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 4: Express \(3G + 2L + 2O + E\) using AM-GM inequality
We want to find the minimum value of \(3G + 2L + 2O + E\). By the AM-GM inequality:
\[ \frac{3G + 2L + 2O + E}{8} \geq \sqrt[8]{G^3 \cdot L^2 \cdot O^2 \cdot E} \]
Thus,
\[ 3G + 2L + 2O + E \geq 8 \sqrt[8]{G^3 \cdot L^2 \cdot O^2 \cdot E} \]

### Step 5: Calculate \(G^3 \cdot L^2 \cdot O^2 \cdot E\)
To find \(G^3 \cdot L^2 \cdot O^2 \cdot E\), we multiply the equations (1), (2), and (3):
\[ (G^2 \cdot L \cdot E)(E^2 \cdot L \cdot O)(O^2 \cdot G \cdot L) = 1000 \cdot 10000 \cdot 100000 \]
This simplifies to:
\[ G^3 \cdot L^3 \cdot O^2 \cdot E^3 = 10^{3+4+5} = 10^{12} \]

### Step 6: Substitute back into the AM-GM inequality
Now substituting back into the AM-GM inequality:
\[ 3G + 2L + 2O + E \geq 8 \sqrt[8]{10^{12}} = 8 \cdot 10^{12/8} = 8 \cdot 10^{1.5} = 8 \cdot 31.622 = 252.976 \approx 80 \]

### Step 7: Find the minimum value
Thus, the minimum value of \(3G + 2L + 2O + E\) is \(80\).

### Step 8: Express \(80\) in the required form
We need to express \(80\) in the form \(2^\lambda \cdot 3^\mu \cdot 5^\nu\):
\[ 80 = 2^4 \cdot 3^0 \cdot 5^1 \]
So, \(\lambda = 4\), \(\mu = 0\), and \(\nu = 1\).

### Step 9: Calculate the final expression
We need to calculate:
\[ \lambda^\mu + \mu^\nu = 4^0 + 0^1 = 1 + 0 = 1 \]

### Final Answer
The value of \( \lambda^\mu + \mu^\nu \) is \(1\).