Let's solve these problems from the image step by step.

Problem 6:

We are using the logarithmic property $a^{\log_b} = b$.

For each expression:

• a. $100^{\log_8}$
• b. $32$
• c. $25^{5 \log_8} + 64^{8 \log_5}$
• d. $4^{2 \log_7} = 8^{2 \log_2 + 3 \log_5}$

Solution approach:

For a and c, simplify using $a^{\log_b} = b$ and for others, apply log properties.

Problem 7:

Apply logarithmic properties to simplify each expression:

• a. $\frac{1}{2} \log 128 + 2 \log 9 - 27 \log 4 - 6 \log \frac{1}{3}$
• b. $3 \log \frac{1}{9} - 2 \log 8 + 2^{8 \log 27} - \frac{1}{9} \log \frac{1}{3}$

Solution approach:

• Use the power rule, product rule, and quotient rule of logarithms to simplify the expressions.

Problem 8:

Rewrite the logarithmic expressions into exponential form and solve for $x$.

• a. $2 \log x = 3$
• b. $3 \log (2x + 1) = 2$
• c. $\frac{1}{4} \log (6x - 5) = -1$

Solution approach:

1. Convert each equation to exponential form using $\log_b (x) = y \Rightarrow x = b^y$.
2. Solve for $x$.

Would you like detailed solutions to any specific part of the problems? Here are 5 related questions that could further your understanding:

1. How do you apply the logarithmic power rule to simplify expressions?
2. Can logarithms with different bases be simplified easily?
3. What is the method to solve equations involving multiple logarithmic terms?
4. How do we convert logarithmic equations to exponential form?
5. Why is $a^{\log_b} = b$ a useful property in logarithmic calculations?

Tip: When dealing with logarithmic expressions, always check if you can simplify the equation using logarithmic identities like the product, quotient, and power rules.