7 log `(10)/(9) + 3 log""(81)/(80)` =

Options

A. 4 log 3 - 2 log 5 - 5 log 2
B. 3 log 4- 5 log 2 - 2 log 5
C. 4 log 5 -2 log 3-5 log 2
D. none of these

Answer

To solve the equation \( 7 \log \frac{10}{9} + 3 \log \frac{81}{80} \), we will break it down step by step using logarithmic properties.

### Step 1: Apply the logarithmic property of fractions
We know that:
\[ \log \frac{m}{n} = \log m - \log n \]
Using this property, we can rewrite the expression:
\[ 7 \log \frac{10}{9} + 3 \log \frac{81}{80} = 7 (\log 10 - \log 9) + 3 (\log 81 - \log 80) \]

### Step 2: Distribute the coefficients
Distributing the coefficients gives us:
\[ = 7 \log 10 - 7 \log 9 + 3 \log 81 - 3 \log 80 \]

### Step 3: Simplify the logarithms
Next, we can simplify the logarithms:
- \( \log 10 = \log(2 \cdot 5) = \log 2 + \log 5 \)
- \( \log 9 = \log(3^2) = 2 \log 3 \)
- \( \log 81 = \log(3^4) = 4 \log 3 \)
- \( \log 80 = \log(2^4 \cdot 5) = 4 \log 2 + \log 5 \)

Substituting these into our expression:
\[ = 7 (\log 2 + \log 5) - 7(2 \log 3) + 3(4 \log 3) - 3(4 \log 2 + \log 5) \]

### Step 4: Expand and combine like terms
Now, we expand and combine like terms:
\[ = 7 \log 2 + 7 \log 5 - 14 \log 3 + 12 \log 3 - 12 \log 2 - 3 \log 5 \]
Combining the terms gives:
\[ = (7 \log 2 - 12 \log 2) + (7 \log 5 - 3 \log 5) + (-14 \log 3 + 12 \log 3) \]
\[ = -5 \log 2 + 4 \log 5 - 2 \log 3 \]

### Step 5: Final expression
The final expression is:
\[ = 4 \log 5 - 2 \log 3 - 5 \log 2 \]

### Conclusion
The simplified form of \( 7 \log \frac{10}{9} + 3 \log \frac{81}{80} \) is:
\[ 4 \log 5 - 2 \log 3 - 5 \log 2 \]