The image contains logarithmic expressions in the context of simplifying equations. Let's break down the question and begin simplifying each expression.

9. Simplify the following logarithmic expressions:

a. $\log^6 3 + \log^6 12$

This appears to involve logarithmic properties, likely focusing on the same base of $\log$. Use properties of logs, such as: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$

b. $\log^6 9 + 2 \cdot \log^6 2 - 2 \cdot \log^6 6$

This combines multiple logarithms, using both addition and subtraction. You can apply the following: $\log_b(x^n) = n \cdot \log_b(x)$ as well as addition and subtraction of logs: $\log_b(x) + \log_b(y) = \log_b(x \cdot y)$ $\log_b(x) - \log_b(y) = \log_b \left( \frac{x}{y} \right)$

c. $\log^2 12 + \log^1 \left( \frac{16}{9} \right)$

Here, we combine two logs, one of a constant and one of a fraction. The properties of logarithms allow us to simplify fractions inside the log: $\log_b \left( \frac{x}{y} \right) = \log_b(x) - \log_b(y)$

d. $\log^5 3x + \log^5 \left( \frac{1}{25} \right)$

This involves multiplication and a fraction within the logarithms, so applying: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ and for fractions: $\log_b(x^n) = n \cdot \log_b(x)$

e. $\log^2 10x^6 \log^4 \times \log 216$

Here, we see multiple logarithms combined with multiplication. The rule to apply is: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ We also use properties involving exponents, such as $x^6$ inside the logarithm.

Let me know if you'd like detailed steps to solve each of these, or if you want to focus on specific ones!

5 Related Questions:

1. How do you combine multiple logarithmic terms using addition or subtraction?
2. What is the property of logarithms when dealing with exponents inside a logarithmic function?
3. How do you simplify a logarithmic expression involving fractions?
4. What happens when you apply the change of base formula to logarithms?
5. How do you apply logarithmic properties in equations with products?

Tip: Always try simplifying logarithmic expressions by applying the logarithm product, quotient, and power rules step by step.