Simplify :
` 2 log ""( 15)/(8) - log ""( 25)/( 162) +3 log ""( 4)/(9)`

Answer

To simplify the expression \( 2 \log \frac{15}{8} - \log \frac{25}{162} + 3 \log \frac{4}{9} \), we can follow these steps:

### Step 1: Rewrite the Logs
We start by rewriting the logarithmic expressions using the properties of logarithms. The property we will use is:
\[ \log \frac{A}{B} = \log A - \log B \]

Using this property, we can rewrite each term:
\[ 2 \log \frac{15}{8} = 2 (\log 15 - \log 8) = 2 \log 15 - 2 \log 8 \]
\[ -\log \frac{25}{162} = -(\log 25 - \log 162) = -\log 25 + \log 162 \]
\[ 3 \log \frac{4}{9} = 3 (\log 4 - \log 9) = 3 \log 4 - 3 \log 9 \]

So the expression becomes:
\[ 2 \log 15 - 2 \log 8 - \log 25 + \log 162 + 3 \log 4 - 3 \log 9 \]

### Step 2: Substitute Known Values
Next, we can express the numbers in terms of their prime factors:
- \( 15 = 3 \times 5 \)
- \( 8 = 2^3 \)
- \( 25 = 5^2 \)
- \( 162 = 2 \times 3^4 \)
- \( 4 = 2^2 \)
- \( 9 = 3^2 \)

Now substituting these values into the logarithm:
\[ 2 \log (3 \times 5) - 2 \log (2^3) - \log (5^2) + \log (2 \times 3^4) + 3 \log (2^2) - 3 \log (3^2) \]

### Step 3: Apply Logarithmic Properties
Using the properties \( \log (AB) = \log A + \log B \) and \( \log (A^n) = n \log A \):
\[ 2 (\log 3 + \log 5) - 2 (3 \log 2) - 2 \log 5 + (\log 2 + 4 \log 3) + 3 (2 \log 2) - 3 (2 \log 3) \]

This simplifies to:
\[ 2 \log 3 + 2 \log 5 - 6 \log 2 - 2 \log 5 + \log 2 + 4 \log 3 + 6 \log 2 - 6 \log 3 \]

### Step 4: Combine Like Terms
Now, we can combine like terms:
- For \( \log 3 \):
\[ (2 \log 3 + 4 \log 3 - 6 \log 3) = 0 \log 3 = 0 \]
- For \( \log 5 \):
\[ (2 \log 5 - 2 \log 5) = 0 \log 5 = 0 \]
- For \( \log 2 \):
\[ (-6 \log 2 + \log 2 + 6 \log 2) = 1 \log 2 = \log 2 \]

### Final Result
Thus, the entire expression simplifies to:
\[ \log 2 \]