The value of `7 "log"_(a)(16)/(15)+5 "log"_(a) (25)/(24) + 3"log"_(a) (81)/(80)` is

Options

A. `log_(a)5`
B. `log_(a) 3`
C. `log_(a) 2`
D. None of these

Answer

To solve the expression \( 7 \log_a \left(\frac{16}{15}\right) + 5 \log_a \left(\frac{25}{24}\right) + 3 \log_a \left(\frac{81}{80}\right) \), we will use the properties of logarithms step by step.

### Step 1: Apply the logarithmic property
Using the property \( \log \left(\frac{m}{n}\right) = \log m - \log n \), we can rewrite each term:

\[ 7 \log_a \left(\frac{16}{15}\right) = 7 \left(\log_a 16 - \log_a 15\right) \]
\[ 5 \log_a \left(\frac{25}{24}\right) = 5 \left(\log_a 25 - \log_a 24\right) \]
\[ 3 \log_a \left(\frac{81}{80}\right) = 3 \left(\log_a 81 - \log_a 80\right) \]

### Step 2: Expand the expression
Now, we can expand the entire expression:

\[ = 7 \log_a 16 - 7 \log_a 15 + 5 \log_a 25 - 5 \log_a 24 + 3 \log_a 81 - 3 \log_a 80 \]

### Step 3: Rewrite the numbers in terms of their prime factors
Next, we rewrite the numbers \( 16, 15, 25, 24, 81, \) and \( 80 \) in terms of their prime factors:

- \( 16 = 2^4 \)
- \( 15 = 3 \times 5 \)
- \( 25 = 5^2 \)
- \( 24 = 2^3 \times 3 \)
- \( 81 = 3^4 \)
- \( 80 = 2^4 \times 5 \)

### Step 4: Substitute the prime factorization into the expression
Now substituting these back into the expression:

\[ = 7 \log_a (2^4) - 7 \log_a (3 \times 5) + 5 \log_a (5^2) - 5 \log_a (2^3 \times 3) + 3 \log_a (3^4) - 3 \log_a (2^4 \times 5) \]

### Step 5: Use the power and product properties of logarithms
Using the properties \( \log_a (m^n) = n \log_a m \) and \( \log_a (mn) = \log_a m + \log_a n \):

\[ = 7 \cdot 4 \log_a 2 - 7 (\log_a 3 + \log_a 5) + 5 \cdot 2 \log_a 5 - 5 (\log_a (2^3) + \log_a 3) + 3 \cdot 4 \log_a 3 - 3 (\log_a (2^4) + \log_a 5) \]

This simplifies to:

\[ = 28 \log_a 2 - 7 \log_a 3 - 7 \log_a 5 + 10 \log_a 5 - 15 \log_a 2 - 5 \log_a 3 + 12 \log_a 3 - 12 \log_a 2 - 3 \log_a 5 \]

### Step 6: Combine like terms
Now, we combine the coefficients of \( \log_a 2, \log_a 3, \) and \( \log_a 5 \):

- Coefficient of \( \log_a 2 \):
\[ 28 - 15 - 12 = 1 \]

- Coefficient of \( \log_a 3 \):
\[ -7 - 5 + 12 = 0 \]

- Coefficient of \( \log_a 5 \):
\[ -7 + 10 - 3 = 0 \]

### Final Result
Thus, we have:

\[ = 1 \log_a 2 + 0 \log_a 3 + 0 \log_a 5 = \log_a 2 \]

The value of the expression is \( \log_a 2 \).