The value of `("log" 49 sqrt(7) + "log" 25sqrt(5) - "log" 4sqrt(2))/("log" 17.5)`, is

Options

A. 5
B. 2
C. `(5)/(2)`
D. `(3)/(2)`

Answer

To solve the expression \(\frac{\log(49\sqrt{7}) + \log(25\sqrt{5}) - \log(4\sqrt{2})}{\log(17.5)}\), we will follow these steps:

### Step 1: Rewrite the logarithmic arguments
We can express \(49\), \(25\), and \(4\) in terms of their bases:
- \(49 = 7^2\)
- \(25 = 5^2\)
- \(4 = 2^2\)

Thus, we can rewrite the logs as:
\[ \log(49\sqrt{7}) = \log(7^2 \cdot 7^{1/2}) = \log(7^{2 + 1/2}) = \log(7^{5/2}) \]
\[ \log(25\sqrt{5}) = \log(5^2 \cdot 5^{1/2}) = \log(5^{2 + 1/2}) = \log(5^{5/2}) \]
\[ \log(4\sqrt{2}) = \log(2^2 \cdot 2^{1/2}) = \log(2^{2 + 1/2}) = \log(2^{5/2}) \]

### Step 2: Substitute back into the expression
Now, substituting back into the original expression, we have:
\[ \frac{\log(7^{5/2}) + \log(5^{5/2}) - \log(2^{5/2})}{\log(17.5)} \]

### Step 3: Use the properties of logarithms
Using the property \(\log(a^b) = b \log(a)\), we can simplify:
\[ \log(7^{5/2}) = \frac{5}{2} \log(7), \quad \log(5^{5/2}) = \frac{5}{2} \log(5), \quad \log(2^{5/2}) = \frac{5}{2} \log(2) \]

Thus, the expression becomes:
\[ \frac{\frac{5}{2} \log(7) + \frac{5}{2} \log(5) - \frac{5}{2} \log(2)}{\log(17.5)} \]

### Step 4: Factor out the common term
We can factor out \(\frac{5}{2}\) from the numerator:
\[ \frac{5}{2} \cdot \frac{\log(7) + \log(5) - \log(2)}{\log(17.5)} \]

### Step 5: Combine the logarithms
Using the properties of logarithms, we can combine the logs in the numerator:
\[ \log(7) + \log(5) - \log(2) = \log\left(\frac{7 \cdot 5}{2}\right) = \log\left(\frac{35}{2}\right) \]

### Step 6: Substitute back into the expression
Now, we substitute this back into the expression:
\[ \frac{5}{2} \cdot \frac{\log\left(\frac{35}{2}\right)}{\log(17.5)} \]

### Step 7: Simplify the fraction
Since \(\frac{35}{2} = 17.5\), we have:
\[ \log\left(\frac{35}{2}\right) = \log(17.5) \]

Thus, the expression simplifies to:
\[ \frac{5}{2} \cdot \frac{\log(17.5)}{\log(17.5)} = \frac{5}{2} \]

### Final Answer
The value of the expression is:
\[ \frac{5}{2} \]