We are given:

• $\log_5 6 = a$
• $\log_5 11 = b$

We need to rewrite these logarithms using the change of base formula and express them in terms of $a$ and $b$.

Change of Base Formula:

The change of base formula states that:

$\log_b x = \frac{\log_c x}{\log_c b}$

Where $c$ is any base (typically base 10 or $e$ for natural logs). However, in this case, we can use the known values of $a$ and $b$ to simplify expressions involving logarithms in base 5.

Expressions to Rewrite:

1. $\log_5 66$

   • By the property of logarithms:

     $\log_5 66 = \log_5 (6 \times 11) = \log_5 6 + \log_5 11$ Using the definitions of $a$ and $b$:

     $\log_5 66 = a + b$

2. $\log_5 \left( \frac{11}{6} \right)$

   • Using the quotient property of logarithms:

     $\log_5 \left( \frac{11}{6} \right) = \log_5 11 - \log_5 6$ Which simplifies to:

     $\log_5 \left( \frac{11}{6} \right) = b - a$

Final Answer:

• $\log_5 66 = a + b$
• $\log_5 \left( \frac{11}{6} \right) = b - a$

Would you like to explore more details or have any other questions?

Here are 5 related questions for practice:

1. What is the value of $\log_5 36$ in terms of $a$?
2. How can you express $\log_5 1$ using properties of logarithms?
3. Use the change of base formula to express $\log_2 6$ in terms of $\log_5 6$ and $\log_5 2$.
4. How do you express $\log_5 (6^2)$ in terms of $a$?
5. How do you express $\log_5 \left( \frac{6^2}{11} \right)$ in terms of $a$ and $b$?

Tip: Logarithm properties like the product, quotient, and power rules are extremely useful for simplifying expressions.