Let's go through each logarithmic expression:

1. For $\log_2 64$:

   The expression $\log_2 64$ asks for the power to which 2 must be raised to get 64. $2^6 = 64$ Therefore, $\log_2 64 = 6$.

2. For $\log_5 \frac{1}{25}$:

   The expression $\log_5 \frac{1}{25}$ asks for the power to which 5 must be raised to get $\frac{1}{25}$. $5^{-2} = \frac{1}{25}$ So, $\log_5 \frac{1}{25} = -2$.

3. For $\log_3 27$:

   The expression $\log_3 27$ asks for the power to which 3 must be raised to get 27. $3^3 = 27$ Therefore, $\log_3 27 = 3$.

4. For $\log_3 3$:

   The expression $\log_3 3$ asks for the power to which 3 must be raised to get 3. $3^1 = 3$ So, $\log_3 3 = 1$.

Here are the answers:

• $\log_2 64 = 6$
• $\log_5 \frac{1}{25} = -2$
• $\log_3 27 = 3$
• $\log_3 3 = 1$

Would you like a further breakdown of any specific problem?

Related Questions:

1. How would you calculate $\log_4 64$?
2. What is the value of $\log_{10} 1000$?
3. Can you solve $\log_6 1$ without using a calculator?
4. What happens if the base and the number are the same in logarithmic form, like $\log_7 7$?
5. How do you express $\log_b \frac{1}{b^k}$ in terms of $k$?

Tip:

When you see a fraction like $\frac{1}{n}$ in a logarithmic expression, think about negative exponents.