We are given the equation \( \log_2 3 = P \) and need to determine the value of \( \log_8 81 \).

### Step 1: Express \( \log_8 81 \) in terms of \( \log_2 3 \)

We can break down both the base 8 and the argument 81 into powers of simpler numbers.

- \( 8 = 2^3 \)
- \( 81 = 3^4 \)

Thus, we rewrite \( \log_8 81 \) as:

\[
\log_8 81 = \log_{2^3} 3^4
\]

### Step 2: Use the change of base formula

The change of base formula states:

\[
\log_b a = \frac{\log_k a}{\log_k b}
\]

In this case, we apply it to \( \log_{2^3} 3^4 \):

\[
\log_{2^3} 3^4 = \frac{\log_2 3^4}{\log_2 2^3}
\]

### Step 3: Simplify the expression

- \( \log_2 3^4 = 4 \log_2 3 \)
- \( \log_2 2^3 = 3 \)

Thus, we have:

\[
\log_8 81 = \frac{4 \log_2 3}{3}
\]

### Step 4: Substitute \( \log_2 3 = P \)

Now substitute \( \log_2 3 = P \):

\[
\log_8 81 = \frac{4P}{3}
\]

### Final Answer:

\[
\log_8 81 = \frac{4P}{3}
\]

Would you like more details or further explanation?

Here are 5 related questions to explore:
1. How can you use the change of base formula in logarithmic calculations?
2. How does the relationship between logarithms and exponents work?
3. Can you generalize \( \log_b a \) if both \( a \) and \( b \) are powers of other numbers?
4. How would you convert between different logarithmic bases?
5. What are the key properties of logarithms?

**Tip:** Always check if you can express both the base and the argument of a logarithm as powers of simpler numbers for easier computation!

Learn how to solve the logarithmic equation log8 81, given that log2 3 = P, using the change of base formula. This solution explores key logarithmic concepts and simplifies the expression with powers of 2 and 3. Suitable for high school students in Grades 10-12.

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