Simplify $\frac{\log{\sqrt{a}}}{\log{a}} \times \frac{\log{\sqrt[3]{5}}}{\log{\sqrt{5}}}$

Steps to Solve

1. Simplify the first fraction

Using the power rule of logarithms, we can simplify the numerator $\log{\sqrt{a}}$ as $\log{a^{\frac{1}{2}}}$ which equals $\frac{1}{2}\log{a}$. Thus, the first fraction becomes: $$ \frac{\log{\sqrt{a}}}{\log{a}} = \frac{\frac{1}{2}\log{a}}{\log{a}} = \frac{1}{2} $$

2. Simplify the second fraction Similarly, we apply the power rule to the second fraction's numerator and denominator: $\log{\sqrt[3]{5}} = \log{5^{\frac{1}{3}}} = \frac{1}{3}\log{5}$ and $\log{\sqrt{5}} = \log{5^{\frac{1}{2}}} = \frac{1}{2}\log{5}$. Thus the second fraction simplifies to: $$ \frac{\log{\sqrt[3]{5}}}{\log{\sqrt{5}}} = \frac{\frac{1}{3}\log{5}}{\frac{1}{2}\log{5}} = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{1}{3} \times \frac{2}{1} = \frac{2}{3} $$

3. Multiply the simplified fractions Now, we multiply the simplified fractions: $$ \frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} $$

4. Further simplification We simplify the resulting fraction to its simplest form: $$ \frac{2}{6} = \frac{1}{3} $$