Simplify the numerator using logarithm properties.

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Here is the solution to the logarithmic expression.

Simplify the numerator using logarithm properties.
The numerator is $\log \sqrt{27} + \log \sqrt{8} - \log \sqrt{125}$ .
First, rewrite the square roots as powers:
$\sqrt{27} = 27^{1/2}$ $\sqrt{8} = 8^{1/2}$ $\sqrt{125} = 125^{1/2}$
Substitute these into the numerator:
$\log (27^{1/2}) + \log (8^{1/2}) - \log (125^{1/2})$
Apply the logarithm power rule, $n \log a = \log a^n$ :
$\frac{1}{2} \log 27 + \frac{1}{2} \log 8 - \frac{1}{2} \log 125$
Factor out $\frac{1}{2}$ :
$\frac{1}{2} (\log 27 + \log 8 - \log 125)$
Rewrite the numbers inside the logarithms as powers of their prime factors:
$27 = 3^3$ $8 = 2^3$ $125 = 5^3$
Substitute these back:
$\frac{1}{2} (\log (3^3) + \log (2^3) - \log (5^3))$
Apply the logarithm power rule again:
$\frac{1}{2} (3 \log 3 + 3 \log 2 - 3 \log 5)$
Factor out 3:
$\frac{1}{2} \cdot 3 (\log 3 + \log 2 - \log 5)$ $\frac{3}{2} (\log 3 + \log 2 - \log 5)$
Apply the logarithm product rule, $\log a + \log b = \log (ab)$ , and the quotient rule, $\log a - \log b = \log \left(\frac{a}{b}\right)$ :
$\frac{3}{2} (\log (3 \times 2) - \log 5)$ $\frac{3}{2} (\log 6 - \log 5)$
Simplify the denominator.
The denominator is $\log 6 - \log 5$ . This is already in a simplified form for comparison.
Combine the simplified numerator and denominator.
The original expression becomes:
$\frac{\frac{3}{2} (\log 6 - \log 5)}{\log 6 - \log 5}$
Since $(\log 6 - \log 5)$ is a common factor in both the numerator and the denominator, and it is not zero (because $\log 6 \neq \log 5$ ), we can cancel it out.
$\frac{3}{2}$

Simplify the numerator using logarithm properties.

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Simplify the numerator using logarithm properties.

Simplify the denominator.

Combine the simplified numerator and denominator.

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