Find the value of `(log sqrt(27) + log sqrt(8) - log sqrt(125))/(log 6 - log 5)`

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 terms in the numerator
Using the property of logarithms that states \(\log a^b = b \log a\), we can rewrite the square roots:

\[ \log \sqrt{27} = \log(27^{1/2}) = \frac{1}{2} \log 27 \]
\[ \log \sqrt{8} = \log(8^{1/2}) = \frac{1}{2} \log 8 \]
\[ \log \sqrt{125} = \log(125^{1/2}) = \frac{1}{2} \log 125 \]

### Step 2: Substitute the simplified logarithms back into the expression
Now substitute these back 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 \(\frac{1}{2}\) from the numerator
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: Use the property of logarithms to combine terms
Using the property \(\log a + \log b = \log(ab)\) and \(\log a - \log b = \log(\frac{a}{b})\), we can rewrite the numerator:

\[ \log 27 + \log 8 - \log 125 = \log\left(\frac{27 \cdot 8}{125}\right) \]

### Step 5: Calculate \(27 \cdot 8\) and simplify
Calculate \(27 \cdot 8\):

\[ 27 \cdot 8 = 216 \]

Thus, the numerator becomes:

\[ \log\left(\frac{216}{125}\right) \]

### Step 6: Simplify the denominator
The denominator can be simplified using the same logarithmic properties:

\[ \log 6 - \log 5 = \log\left(\frac{6}{5}\right) \]

### Step 7: Substitute back into the expression
Now substitute back into the expression:

\[ = \frac{1}{2} \cdot \frac{\log\left(\frac{216}{125}\right)}{\log\left(\frac{6}{5}\right)} \]

### Step 8: Evaluate the final expression
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 values
Now we can compute the values:

- \(216 = 6^3\)
- \(125 = 5^3\)

Thus:

\[ \frac{216}{125} = \frac{6^3}{5^3} = \left(\frac{6}{5}\right)^3 \]

So:

\[ \log_{\frac{6}{5}}\left(\frac{216}{125}\right) = \log_{\frac{6}{5}}\left(\left(\frac{6}{5}\right)^3\right) = 3 \]

### Step 10: Final answer
Now substitute this back into our expression:

\[ = \frac{1}{2} \cdot 3 = \frac{3}{2} \]

Thus, the final value is:

\[ \frac{3}{2} \]

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