Find the value of `"log"(9)/(8)-"log"(27)/(32)+"log"(3)/(4)`

Options

A. A)0
B. B)1
C. C)3
D. D)`log(3//4)`

Answer

To solve the expression \(\frac{\log(9)}{8} - \frac{\log(27)}{32} + \frac{\log(3)}{4}\), we can use the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Rewrite the logarithmic terms
We can rewrite the logarithmic terms to have a common base. The expression can be rewritten as:
\[ \log(9) = \log(3^2) = 2\log(3) \]
\[ \log(27) = \log(3^3) = 3\log(3) \]

### Step 2: Substitute the rewritten logarithms into the expression
Substituting these into the original expression gives:
\[ \frac{2\log(3)}{8} - \frac{3\log(3)}{32} + \frac{\log(3)}{4} \]

### Step 3: Simplify each term
Now simplify each term:
\[ \frac{2\log(3)}{8} = \frac{\log(3)}{4} \]
So, the expression becomes:
\[ \frac{\log(3)}{4} - \frac{3\log(3)}{32} + \frac{\log(3)}{4} \]

### Step 4: Combine like terms
Now, combine the terms:
\[ \frac{\log(3)}{4} + \frac{\log(3)}{4} = \frac{2\log(3)}{4} = \frac{\log(3)}{2} \]
Thus, the expression now is:
\[ \frac{\log(3)}{2} - \frac{3\log(3)}{32} \]

### Step 5: Find a common denominator
The common denominator for \(2\) and \(32\) is \(32\). Rewrite \(\frac{\log(3)}{2}\) as:
\[ \frac{\log(3)}{2} = \frac{16\log(3)}{32} \]

### Step 6: Combine the fractions
Now, combine the fractions:
\[ \frac{16\log(3)}{32} - \frac{3\log(3)}{32} = \frac{(16 - 3)\log(3)}{32} = \frac{13\log(3)}{32} \]

### Step 7: Final expression
Thus, the final expression simplifies to:
\[ \frac{13\log(3)}{32} \]

### Conclusion
The value of the original expression \(\frac{\log(9)}{8} - \frac{\log(27)}{32} + \frac{\log(3)}{4}\) is \(\frac{13\log(3)}{32}\).