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

Options

A. 0.3.10
B. 0.3512
C. 0.412
D. None of these

Answer

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

### Step 1: Combine the logarithmic expressions
Using the property \( a \log m + b \log n = \log(m^a) + \log(n^b) = \log(m^a \cdot n^b) \), we can combine the logarithmic terms:

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

### Step 2: Calculate the powers
Now we will calculate each term inside the logarithm:

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

### Step 3: Substitute back into the logarithm
Substituting these values back, we have:

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

### Step 4: Simplify the numerator and denominator
Next, we can simplify the numerator and denominator:

- \( 16 = 4^2 \) so \( 16^7 = (4^2)^7 = 4^{14} \)
- \( 25 = 5^2 \) so \( 25^5 = (5^2)^5 = 5^{10} \)
- \( 81 = 3^4 \) so \( 81^3 = (3^4)^3 = 3^{12} \)

Thus, the numerator becomes:

\[ 4^{14} \cdot 5^{10} \cdot 3^{12} \]

For the denominator:

- \( 15 = 3 \cdot 5 \) so \( 15^7 = (3 \cdot 5)^7 = 3^7 \cdot 5^7 \)
- \( 24 = 3 \cdot 8 = 3 \cdot 2^3 \) so \( 24^5 = (3 \cdot 2^3)^5 = 3^5 \cdot 2^{15} \)
- \( 80 = 16 \cdot 5 = 4^2 \cdot 5 \) so \( 80^3 = (4^2 \cdot 5)^3 = 4^6 \cdot 5^3 \)

Thus, the denominator becomes:

\[ 3^7 \cdot 5^7 \cdot 3^5 \cdot 2^{15} \cdot 4^6 \cdot 5^3 = 3^{12} \cdot 5^{10} \cdot 2^{15} \cdot 4^6 \]

### Step 5: Combine and simplify
Now we can combine the numerator and denominator:

\[ \log \left( \frac{4^{14} \cdot 5^{10} \cdot 3^{12}}{3^{12} \cdot 5^{10} \cdot 2^{15} \cdot 4^6} \right) \]

This simplifies to:

\[ \log \left( \frac{4^{14 - 6}}{2^{15}} \right) = \log \left( \frac{4^8}{2^{15}} \right) \]

Since \( 4 = 2^2 \), we have:

\[ \log \left( \frac{(2^2)^8}{2^{15}} \right) = \log \left( \frac{2^{16}}{2^{15}} \right) = \log(2) \]

### Final Answer
Thus, the final value of the expression is:

\[ \log(2) \]