If m = log 20 and n = log 25, find the value of x, so that : 2 log(x - 4) = 2m - n.

Answer

To solve the equation \(2 \log(x - 4) = 2m - n\) given that \(m = \log 20\) and \(n = \log 25\), we will follow these steps:

### Step 1: Substitute the values of \(m\) and \(n\)
We know:
- \(m = \log 20\)
- \(n = \log 25\)

Substituting these into the equation gives:
\[ 2 \log(x - 4) = 2 \log 20 - \log 25 \]

### Step 2: Simplify the right side
Using the properties of logarithms, we can rewrite \(2 \log 20\) as \(\log(20^2)\) and \(n\) as \(\log 25\):
\[ 2 \log 20 = \log(20^2) = \log 400 \]
Thus, we have:
\[ 2 \log(x - 4) = \log 400 - \log 25 \]

### Step 3: Apply the quotient rule of logarithms
Using the property \(\log a - \log b = \log\left(\frac{a}{b}\right)\), we can simplify the right side:
\[ \log 400 - \log 25 = \log\left(\frac{400}{25}\right) \]
Calculating \(\frac{400}{25}\):
\[ \frac{400}{25} = 16 \]
So we have:
\[ 2 \log(x - 4) = \log 16 \]

### Step 4: Simplify the left side
We can rewrite \(2 \log(x - 4)\) as \(\log((x - 4)^2)\):
\[ \log((x - 4)^2) = \log 16 \]

### Step 5: Set the arguments equal to each other
Since the logarithms are equal, we can set the arguments equal:
\[ (x - 4)^2 = 16 \]

### Step 6: Solve for \(x\)
Taking the square root of both sides gives:
\[ x - 4 = 4 \quad \text{or} \quad x - 4 = -4 \]
From \(x - 4 = 4\):
\[ x = 8 \]
From \(x - 4 = -4\):
\[ x = 0 \]
However, since \(x - 4\) must be positive (as it is inside a logarithm), we discard \(x = 0\).

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{8} \]