Find the value of `(logsqrt27+logsqrt8-logsqrt125)/(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 can follow these steps:

### Step 1: Simplify the logarithmic expressions
We start by rewriting the square roots in terms of logarithms:
\[ \log \sqrt{27} = \log(27^{1/2}) = \frac{1}{2} \log 27 \]
\[ \log \sqrt{8} = \log(8^{1/2}) = \frac{1}{2} \log 8 \]
\[ \log \sqrt{125} = \log(125^{1/2}) = \frac{1}{2} \log 125 \]

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

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

### Step 4: Use the properties of logarithms
Using the property \(\log a + \log b = \log(ab)\) and \(\log a - \log b = \log\left(\frac{a}{b}\right)\):
\[ = \frac{\frac{1}{2} \left(\log(27 \cdot 8) - \log 125\right)}{\log\left(\frac{6}{5}\right)} \]
\[ = \frac{\frac{1}{2} \log\left(\frac{27 \cdot 8}{125}\right)}{\log\left(\frac{6}{5}\right)} \]

### Step 5: Calculate \(27 \cdot 8\) and simplify
Calculating \(27 \cdot 8\):
\[ 27 \cdot 8 = 216 \]
Now substituting back:
\[ = \frac{\frac{1}{2} \log\left(\frac{216}{125}\right)}{\log\left(\frac{6}{5}\right)} \]

### Step 6: Simplify \(\frac{216}{125}\)
We can express \(216\) and \(125\) as powers:
\[ 216 = 6^3 \quad \text{and} \quad 125 = 5^3 \]
Thus:
\[ \frac{216}{125} = \frac{6^3}{5^3} = \left(\frac{6}{5}\right)^3 \]
So:
\[ \log\left(\frac{216}{125}\right) = \log\left(\left(\frac{6}{5}\right)^3\right) = 3 \log\left(\frac{6}{5}\right) \]

### Step 7: Substitute back into the expression
Now substituting this back:
\[ = \frac{\frac{1}{2} \cdot 3 \log\left(\frac{6}{5}\right)}{\log\left(\frac{6}{5}\right)} \]
The \(\log\left(\frac{6}{5}\right)\) cancels out:
\[ = \frac{3}{2} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{3}{2} \]