Let's solve each of the logarithmic expressions presented in the image:

Problem (f)

$\log_5 \frac{5^{\frac{3}{5}}}{5} = \log_5 \frac{5^{\frac{3}{5}}}{5^1} = \log_5 5^{\frac{3}{5} - 1} = \log_5 5^{-\frac{2}{5}} = -\frac{2}{5}$

Problem (g)

$\log_6 36^{\frac{4}{6}} = \log_6 36^{\frac{2}{3}}$ Using $36 = 6^2$: $= \log_6 (6^2)^{\frac{2}{3}} = \log_6 6^{\frac{4}{3}} = \frac{4}{3}$

Problem (h)

$\log \frac{1000}{\sqrt{10}} = \log \frac{10^3}{10^{\frac{1}{2}}} = \log 10^{3 - \frac{1}{2}} = \log 10^{\frac{5}{2}} = \frac{5}{2}$

Problem (i)

$\log_7 \frac{\sqrt{7}}{49} = \log_7 \frac{7^{\frac{1}{2}}}{7^2} = \log_7 7^{\frac{1}{2} - 2} = \log_7 7^{-\frac{3}{2}} = -\frac{3}{2}$

Let me know if you need any further explanations or have questions!

Here are five related questions to deepen your understanding:

1. How do logarithmic properties simplify expressions with roots and exponents?
2. What steps are useful when simplifying logarithms with fractional exponents?
3. How does the change of base formula work for complex bases?
4. Why do terms simplify when we use $\log_b b^x = x$?
5. How can logarithmic properties apply to real-world exponential growth?

Tip: Always look for ways to express terms in a logarithmic equation in the same base, which often allows for immediate simplification using properties of logarithms.