To evaluate $\log_9 27$ without a calculator, we can express both 9 and 27 as powers of the same base to simplify the logarithm.

Step 1: Rewrite 9 and 27 as powers of 3

1. $9 = 3^2$
2. $27 = 3^3$

So, we can rewrite the logarithm as: $\log_9 27 = \log_{3^2} 3^3$

Step 2: Use the change of base formula

Using the property $\log_{a^m} b^n = \frac{n}{m} \log_a b$, we can simplify as follows: $\log_{3^2} 3^3 = \frac{3}{2} \log_3 3$

Step 3: Simplify $\log_3 3$

Since $\log_3 3 = 1$, we get: $\frac{3}{2} \cdot 1 = \frac{3}{2}$

Final Answer

$\log_9 27 = \frac{3}{2}$

Would you like more details or have any questions?

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Here are some related questions to explore further:

1. How do logarithmic properties help simplify complex expressions?
2. What other methods can be used to evaluate logarithms without a calculator?
3. How does changing the base of a logarithm affect its value?
4. Can you express $\log_{27} 9$ and evaluate it?
5. What happens if you evaluate $\log_9 (1/27)$ instead?

Tip: Remember, expressing numbers as powers of a common base is a powerful strategy for simplifying logarithmic expressions!