The problem consists of two logarithmic expressions that need to be evaluated.

(a) $\log_6 36$

We want to find $\log_6 36$, which is asking, "What power must 6 be raised to in order to get 36?"

Since: $6^2 = 36$ We have: $\log_6 36 = 2$

(b) $\log_8 \frac{1}{64}$

This asks for the logarithm base 8 of $\frac{1}{64}$, i.e., "What power must 8 be raised to in order to get $\frac{1}{64}$?"

We know: $8^2 = 64 \quad \text{so} \quad 8^{-2} = \frac{1}{64}$ Therefore: $\log_8 \frac{1}{64} = -2$

Final Answers:

• (a) $\log_6 36 = 2$
• (b) $\log_8 \frac{1}{64} = -2$

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do you solve logarithmic equations with different bases?
2. What are the properties of logarithms that can simplify their evaluation?
3. How would you express $\log_b 1$ for any base $b$?
4. Can logarithmic expressions with different bases be easily converted into a single base?
5. How do logarithms relate to exponential functions?

Tip: Always remember that $\log_b b^x = x$.