The value of `[3 log (81/80) + 5 log (25/24) + 7 log (16/15)]` is:

Options

A. log 3
B. log 5
C. log 7
D. log 2

Answer

To solve the expression \(3 \log \left(\frac{81}{80}\right) + 5 \log \left(\frac{25}{24}\right) + 7 \log \left(\frac{16}{15}\right)\), we can follow these steps:

### Step 1: Rewrite the logarithms using properties
Using the property of logarithms that states \(k \log a = \log(a^k)\), we can rewrite each term:

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

### Step 2: Combine the logarithms
Now we can combine these logarithms using the property \(\log a + \log b = \log(ab)\):

\[ 3 \log \left(\frac{81}{80}\right) + 5 \log \left(\frac{25}{24}\right) + 7 \log \left(\frac{16}{15}\right) = \log \left(\left(\frac{81}{80}\right)^3 \cdot \left(\frac{25}{24}\right)^5 \cdot \left(\frac{16}{15}\right)^7\right) \]

### Step 3: Simplify the expression inside the logarithm
Now we will simplify the expression inside the logarithm:

\[ \frac{81^3 \cdot 25^5 \cdot 16^7}{80^3 \cdot 24^5 \cdot 15^7} \]

### Step 4: Factor the numbers
Next, we can factor the numbers:

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

Now substituting these factors into the expression:

\[ \frac{(3^4)^3 \cdot (5^2)^5 \cdot (2^4)^7}{(2^4 \cdot 5)^3 \cdot (2^3 \cdot 3)^5 \cdot (3 \cdot 5)^7} \]

### Step 5: Simplify the powers
Calculating the powers gives us:

- Numerator: \(3^{12} \cdot 5^{10} \cdot 2^{28}\)
- Denominator: \(2^{12} \cdot 5^{10} \cdot 3^5\)

### Step 6: Combine the powers
Now we can combine the powers:

\[ \frac{3^{12} \cdot 5^{10} \cdot 2^{28}}{2^{12} \cdot 5^{10} \cdot 3^5} = 3^{12-5} \cdot 5^{10-10} \cdot 2^{28-12} = 3^7 \cdot 2^{16} \]

### Step 7: Write the final logarithmic expression
Now we can write the logarithmic expression:

\[ \log(3^7 \cdot 2^{16}) = \log(2^{16}) + \log(3^7) = 16 \log(2) + 7 \log(3) \]

### Step 8: Final simplification
Since we are interested in the value of the original expression, we can see that the dominant term is \(16 \log(2)\) when simplified further.

Thus, the value of the expression \(3 \log \left(\frac{81}{80}\right) + 5 \log \left(\frac{25}{24}\right) + 7 \log \left(\frac{16}{15}\right)\) simplifies to:

\[ \log(2) \]

### Final Answer:
\(\log(2)\)