I see that you've uploaded a picture containing logarithmic exercises. Let's break down the problems one by one.

Problem 4:

Given that $4 \log 27 = a$, express the following:

• (a) $3 \log 2$
• (b) $\sqrt{2^{\log \frac{1}{9}}}$

Problem 5:

Given that $5 \log 2 = a$ and $2 \log 3 = b$, express:

• (a) $6 \log 50$
• (b) $45 \log 10$

Problem 6:

Determine the value of:

• (a) $\left(2\right)^{2\log 5}$
• (b) $\left(4\right)^{2\log 3}$
• (c) $\left(3\right)^{9\log 5}$
• (d) $\left(\frac{1}{5}\right)^{25\log 4}$
• (e) $3^{\log 5} \cdot 5^{\log \frac{1}{9}}$
• (f) $9^{\log \frac{1}{5}} \cdot 25^{\log 27}$
• (g) $a^{\log \frac{1}{b}} \cdot c^{\log \sqrt{a}} \cdot b^{2\log \frac{1}{c}}$

Problem 7:

Determine the value of:

• (a) $3^{\log \frac{1}{9}} + 3^{\log 27} \cdot 2^{\log 4}$
• (b) $3^{\log 9} + 125^{\log 25}$
• (c) $4^{\log 8} + 4^{\log 2}$

Do you want to go through the solutions for all of these, or is there a specific one you'd like to focus on first?

Here are five related questions to deepen your understanding:

1. How do you convert logarithmic expressions using the properties of logarithms?
2. How can you simplify logarithmic equations involving exponents?
3. What is the change of base formula, and how does it apply to the given problems?
4. How do you simplify expressions like $a^{\log_b c}$?
5. What is the logarithmic identity for $\log_b 1$ and how can it be used in problem-solving?

Tip: When dealing with logarithmic and exponential expressions together, always check if the logarithmic base and exponent base match—it often simplifies the calculation greatly.