Find the value of `(logsqrt(27)+logsqrt(8)-logsqrt(125))/(log6-log5)`

Options

A. `2/3`
B. `1/3`
C. `3/2`
D. None of these

Answer

To solve the expression \((\log \sqrt{27} + \log \sqrt{8} - \log \sqrt{125}) / (\log 6 - \log 5)\), we will follow these steps:

### Step 1: Rewrite the logarithmic terms
Using the property of logarithms, \(\log a^b = b \log a\), we can rewrite each logarithm:
\[ \log \sqrt{27} = \log 27^{1/2} = \frac{1}{2} \log 27 \]
\[ \log \sqrt{8} = \log 8^{1/2} = \frac{1}{2} \log 8 \]
\[ \log \sqrt{125} = \log 125^{1/2} = \frac{1}{2} \log 125 \]

### Step 2: Substitute back into the expression
Substituting these back into the original expression gives:
\[ \frac{\frac{1}{2} \log 27 + \frac{1}{2} \log 8 - \frac{1}{2} \log 125}{\log 6 - \log 5} \]

### Step 3: Factor out \(\frac{1}{2}\)
We can factor out \(\frac{1}{2}\) from the numerator:
\[ = \frac{1}{2} \cdot \frac{\log 27 + \log 8 - \log 125}{\log 6 - \log 5} \]

### Step 4: Combine the logarithms in the numerator
Using the property \(\log a + \log b = \log(ab)\) and \(\log a - \log b = \log\left(\frac{a}{b}\right)\), we can combine the logarithms:
\[ \log 27 + \log 8 - \log 125 = \log\left(\frac{27 \cdot 8}{125}\right) \]

### Step 5: Calculate \(27 \cdot 8\) and simplify
Calculating \(27 \cdot 8\):
\[ 27 \cdot 8 = 216 \]
Thus, we have:
\[ \log\left(\frac{216}{125}\right) \]

### Step 6: Rewrite the denominator
The denominator can be rewritten as:
\[ \log 6 - \log 5 = \log\left(\frac{6}{5}\right) \]

### Step 7: Substitute back into the expression
Now substituting these back, we have:
\[ = \frac{1}{2} \cdot \frac{\log\left(\frac{216}{125}\right)}{\log\left(\frac{6}{5}\right)} \]

### Step 8: Use the change of base formula
Using the change of base formula, we can express this as:
\[ = \frac{1}{2} \cdot \log_{\frac{6}{5}}\left(\frac{216}{125}\right) \]

### Step 9: Calculate the value
Now we need to find the exponent \(x\) such that:
\[ \left(\frac{6}{5}\right)^x = \frac{216}{125} \]
We can see that:
\[ \frac{216}{125} = \left(\frac{6}{5}\right)^3 \]
Thus, \(x = 3\).

### Step 10: Final result
Therefore, substituting back gives:
\[ = \frac{1}{2} \cdot 3 = \frac{3}{2} \]

### Conclusion
The final value is:
\[ \frac{3}{2} \]