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!