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evaluate the following logarithmic functions

Let's evaluate each logarithmic function step-by-step:

1. \( f(x) = \log_2 4 \)
   - We want \( x \) such that \( 2^x = 4 \).
   - Since \( 4 = 2^2 \), \( x = 2 \).
   - **Answer**: \( f(x) = 2 \).

2. \( f(x) = \log_{25} 5 \)
   - We want \( x \) such that \( 25^x = 5 \).
   - Since \( 25 = 5^2 \), \( 5 = 25^{1/2} \).
   - So, \( x = \frac{1}{2} \).
   - **Answer**: \( f(x) = \frac{1}{2} \).

3. \( f(x) = \log_{64} 64 \)
   - We want \( x \) such that \( 64^x = 64 \).
   - \( x = 1 \) since \( 64^1 = 64 \).
   - **Answer**: \( f(x) = 1 \).

4. \( f(x) = \log_{1/3} \frac{1}{9} \)
   - We want \( x \) such that \( \left(\frac{1}{3}\right)^x = \frac{1}{9} \).
   - Since \( \frac{1}{9} = \left(\frac{1}{3}\right)^2 \), \( x = 2 \).
   - **Answer**: \( f(x) = 2 \).

5. \( f(x) = \log_{1/3} 1 \)
   - We want \( x \) such that \( \left(\frac{1}{3}\right)^x = 1 \).
   - Any number to the power of zero is 1, so \( x = 0 \).
   - **Answer**: \( f(x) = 0 \).

6. \( f(x) = \log_{1/3} 27 \)
   - We want \( x \) such that \( \left(\frac{1}{3}\right)^x = 27 \).
   - \( 27 = 3^3 \), so \( \frac{1}{3} \) needs to be raised to \( -3 \) to equal 27.
   - \( x = -3 \).
   - **Answer**: \( f(x) = -3 \).

7. \( f(x) = \log_{16} 4 \)
   - We want \( x \) such that \( 16^x = 4 \).
   - Since \( 16 = 4^2 \), we can express \( 16^x = (4^2)^x = 4^{2x} \).
   - Thus, \( 4^{2x} = 4 \) implies \( 2x = 1 \), so \( x = \frac{1}{2} \).
   - **Answer**: \( f(x) = \frac{1}{2} \).

8. \( f(x) = \log_{1/3} \frac{1}{3} \)
   - We want \( x \) such that \( \left(\frac{1}{3}\right)^x = \frac{1}{3} \).
   - \( x = 1 \) since \( \left(\frac{1}{3}\right)^1 = \frac{1}{3} \).
   - **Answer**: \( f(x) = 1 \).

9. \( f(x) = \log_2 \frac{1}{8} \)
   - We want \( x \) such that \( 2^x = \frac{1}{8} \).
   - Since \( \frac{1}{8} = 2^{-3} \), \( x = -3 \).
   - **Answer**: \( f(x) = -3 \).

10. \( f(x) = \log_{10} 10 \)
    - We want \( x \) such that \( 10^x = 10 \).
    - \( x = 1 \) since \( 10^1 = 10 \).
    - **Answer**: \( f(x) = 1 \).

Would you like further details on any of these solutions, or do you have any questions?

Here are some related questions:
1. How would you solve logarithmic equations involving irrational bases?
2. Can you explain the change of base formula and its applications?
3. How does the concept of logarithms extend to complex numbers?
4. What are some real-world applications of logarithmic functions?
5. How do logarithmic functions relate to exponential growth and decay?

**Tip**: When dealing with fractional bases, remember that negative exponents invert the base.

Evaluate the following logarithmic functions.

Learn how to evaluate logarithmic functions with various bases in this detailed guide. Each solution includes step-by-step explanations, covering important logarithmic properties and exponents. Suitable for high school students.

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