Given ` 3 (log 5 - log 3 ) - ( log 5 - 3 log 6 ) = 2- log m, ` find m.

Answer

To solve the equation \( 3 (\log 5 - \log 3) - (\log 5 - 3 \log 6) = 2 - \log m \), we will follow these steps:

### Step 1: Expand the left-hand side
We start by distributing the 3 in the first term:

\[ 3 (\log 5 - \log 3) = 3 \log 5 - 3 \log 3 \]

So, the equation becomes:

\[ 3 \log 5 - 3 \log 3 - (\log 5 - 3 \log 6) = 2 - \log m \]

### Step 2: Simplify the left-hand side
Now, we can simplify the left-hand side further:

\[ 3 \log 5 - 3 \log 3 - \log 5 + 3 \log 6 = 2 - \log m \]

Combine like terms:

\[ (3 \log 5 - \log 5) + 3 \log 6 - 3 \log 3 = 2 - \log m \]

This simplifies to:

\[ 2 \log 5 + 3 \log 6 - 3 \log 3 = 2 - \log m \]

### Step 3: Rewrite \( \log 6 \)
We can express \( \log 6 \) as \( \log (3 \times 2) \):

\[ \log 6 = \log 3 + \log 2 \]

Substituting this back into the equation gives:

\[ 2 \log 5 + 3 (\log 3 + \log 2) - 3 \log 3 = 2 - \log m \]

### Step 4: Combine the logs
Now, we can simplify further:

\[ 2 \log 5 + 3 \log 3 + 3 \log 2 - 3 \log 3 = 2 - \log m \]

This simplifies to:

\[ 2 \log 5 + 3 \log 2 = 2 - \log m \]

### Step 5: Move \( \log m \) to the left side
Now, we can move \( \log m \) to the left side:

\[ 2 \log 5 + 3 \log 2 + \log m = 2 \]

### Step 6: Combine the logs
Using the property \( \log a + \log b = \log(ab) \), we can combine the logs:

\[ \log (5^2) + \log (2^3) + \log m = \log (5^2 \cdot 2^3 \cdot m) = 2 \]

### Step 7: Rewrite the equation
This gives us:

\[ \log (25 \cdot 8 \cdot m) = 2 \]

### Step 8: Exponentiate both sides
Taking the antilogarithm of both sides, we have:

\[ 25 \cdot 8 \cdot m = 10^2 \]

### Step 9: Solve for \( m \)
This simplifies to:

\[ 200m = 100 \]

Dividing both sides by 200:

\[ m = \frac{100}{200} = \frac{1}{2} \]

Thus, the value of \( m \) is:

\[ \boxed{\frac{1}{2}} \]