https://file.aiquickdraw.com/mathbot/file/1731006712084slcwtuhm.jpg

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.

Evaluate each expression:
1) \( \log_2 64 \)
2) \( \log_5 \frac{1}{25} \)
3) \( \log_3 27 \)
4) \( \log_3 3 \)

Learn how to evaluate logarithmic expressions with bases 2, 5, and 3. Step-by-step solutions cover expressions such as \( \log_2 64 \), \( \log_5 \frac{1}{25} \), \( \log_3 27 \), and \( \log_3 3 \). Ideal for high school algebra students.

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