The image contains six logarithmic expressions. Let's solve each of them:

1. $\frac{1}{5} \log \sqrt[3]{5} = ?$

Using the logarithmic property $\log(a^b) = b \log(a)$, we first express the cube root as an exponent: $\sqrt[3]{5} = 5^{1/3}$ Now, apply the logarithmic rule: $\log(5^{1/3}) = \frac{1}{3} \log 5$ Thus: $\frac{1}{5} \log \sqrt[3]{5} = \frac{1}{5} \times \frac{1}{3} \log 5 = \frac{1}{15} \log 5$

2. $\sqrt[3]{\log 9} = ?$

Cube root can be expressed as: $\sqrt[3]{\log 9} = (\log 9)^{1/3}$

3. $0.1 \log 10 = ?$

Since $\log 10 = 1$ (for a base-10 logarithm), this simplifies to: $0.1 \log 10 = 0.1 \times 1 = 0.1$

4. $\frac{1}{9} \log \frac{1}{\sqrt[4]{27}} = ?$

First, express the fourth root as an exponent: $\sqrt[4]{27} = 27^{1/4}$ Now, apply the logarithmic property: $\log \frac{1}{27^{1/4}} = \log(27^{-1/4}) = -\frac{1}{4} \log 27$ Thus: $\frac{1}{9} \log \frac{1}{\sqrt[4]{27}} = \frac{1}{9} \times -\frac{1}{4} \log 27 = -\frac{1}{36} \log 27$

5. $0.25 \log \sqrt[3]{2} = ?$

Express the cube root as an exponent: $\log \sqrt[3]{2} = \log 2^{1/3} = \frac{1}{3} \log 2$ Thus: $0.25 \log \sqrt[3]{2} = 0.25 \times \frac{1}{3} \log 2 = \frac{1}{12} \log 2$

6. $\frac{1}{5} \log 0.008 = ?$

We know that $0.008 = 8 \times 10^{-3}$, and $\log(abc) = \log a + \log b + \log c$. Applying this: $\log 0.008 = \log(8 \times 10^{-3}) = \log 8 + \log 10^{-3} = \log 8 - 3$ Thus: $\frac{1}{5} \log 0.008 = \frac{1}{5} (\log 8 - 3) = \frac{1}{5} \log 8 - \frac{3}{5}$

Would you like further clarification on any part?

Additional Questions:

1. How does the logarithmic property $\log(a^b) = b \log(a)$ work in general?
2. Can you explain how to simplify expressions involving logarithms and roots?
3. What are some common logarithmic identities used for simplification?
4. How would changing the base of the logarithm affect these problems?
5. How do you apply logarithmic properties when dealing with negative or fractional inputs?

Tip: Always check if a logarithmic expression can be simplified using exponent properties before calculating values directly.