The simplified form of
`(log ""(75)/(16) -2 log""(5)/(9) + log ""(32)/(243))` is :

Options

A. `log 2`
B. `2 log 2`
C. `log 3`
D. `log 5`

Answer

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

### Step 1: Apply the properties of logarithms
We start with the expression:
\[ \log \left( \frac{75}{16} \right) - 2 \log \left( \frac{5}{9} \right) + \log \left( \frac{32}{243} \right) \]
Using the property \( n \log(a) = \log(a^n) \), we can rewrite \( -2 \log \left( \frac{5}{9} \right) \) as:
\[ - \log \left( \left( \frac{5}{9} \right)^2 \right) \]
Thus, the expression becomes:
\[ \log \left( \frac{75}{16} \right) - \log \left( \left( \frac{5}{9} \right)^2 \right) + \log \left( \frac{32}{243} \right) \]

### Step 2: Combine the logarithms
Using the property \( \log(a) - \log(b) = \log \left( \frac{a}{b} \right) \), we can combine the first two logarithms:
\[ \log \left( \frac{75}{16} \cdot \frac{9^2}{5^2} \right) + \log \left( \frac{32}{243} \right) \]
This simplifies to:
\[ \log \left( \frac{75 \cdot 81}{16 \cdot 25} \right) + \log \left( \frac{32}{243} \right) \]

### Step 3: Further simplify
Now we can combine the two logarithms:
\[ \log \left( \frac{75 \cdot 81 \cdot 32}{16 \cdot 25 \cdot 243} \right) \]

### Step 4: Calculate the values
Calculating the numerator:
\[ 75 \cdot 81 \cdot 32 = 75 \cdot 32 \cdot 81 = 2400 \cdot 81 = 194400 \]
Calculating the denominator:
\[ 16 \cdot 25 \cdot 243 = 400 \cdot 243 = 97200 \]

### Step 5: Final simplification
Now we have:
\[ \log \left( \frac{194400}{97200} \right) \]
Calculating the fraction:
\[ \frac{194400}{97200} = 2 \]

### Final Answer
Thus, we have:
\[ \log(2) \]