Simplify :
` log""(75)/( 16) - 2 log ""( 5)/(9) +log""( 32)/(243)`

Answer

To simplify the expression \( \frac{\log 75}{16} - 2 \frac{\log 5}{9} + \frac{\log 32}{243} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\log 75}{16} - 2 \frac{\log 5}{9} + \frac{\log 32}{243} \]

### Step 2: Use the properties of logarithms
Using the property \( \log \frac{a}{b} = \log a - \log b \), we can rewrite each term:
\[ \log 75 - \log 16 - 2 (\log 5 - \log 9) + \log 32 - \log 243 \]

### Step 3: Expand the expression
Now, we can expand the expression:
\[ \log 75 - \log 16 - 2 \log 5 + 2 \log 9 + \log 32 - \log 243 \]

### Step 4: Substitute known values
Next, we can express \( \log 75 \), \( \log 16 \), \( \log 32 \), and \( \log 243 \) in terms of their prime factors:
- \( \log 75 = \log(25 \cdot 3) = \log(5^2 \cdot 3) = 2 \log 5 + \log 3 \)
- \( \log 16 = \log(4^2) = 2 \log 4 = 2 \cdot 2 \log 2 = 4 \log 2 \)
- \( \log 32 = \log(2^5) = 5 \log 2 \)
- \( \log 243 = \log(3^5) = 5 \log 3 \)

Substituting these into our expression gives:
\[ (2 \log 5 + \log 3) - 4 \log 2 - 2 \log 5 + 2 \log 9 + 5 \log 2 - 5 \log 3 \]

### Step 5: Combine like terms
Now we combine the terms:
- For \( \log 5 \): \( 2 \log 5 - 2 \log 5 = 0 \)
- For \( \log 3 \): \( \log 3 + 2 \log 9 - 5 \log 3 = \log 3 + 4 \log 3 - 5 \log 3 = 0 \)
- For \( \log 2 \): \( -4 \log 2 + 5 \log 2 = 1 \log 2 \)

### Step 6: Final result
The simplified expression is:
\[ \log 2 \]