We want to simplify the expression $\log_2(3) \cdot \log_9(4)$.

Step 1: Change of Base Formula

We can use the change of base formula for logarithms, which states that:

$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$

where $c$ is any positive number (typically we use base 10 or base $e$).

Step 2: Apply the Change of Base Formula to $\log_9(4)$

First, express $\log_9(4)$ in terms of base 3 (since 9 is $3^2$):

$\log_9(4) = \frac{\log_3(4)}{\log_3(9)} = \frac{\log_3(4)}{2}$

Step 3: Substitute and Simplify

Now substitute this into the original expression:

$\log_2(3) \cdot \log_9(4) = \log_2(3) \cdot \frac{\log_3(4)}{2}$

So the expression becomes:

$\frac{1}{2} \cdot \log_2(3) \cdot \log_3(4)$

Step 4: Use Change of Base Formula Again for $\log_2(3)$

Next, express $\log_2(3)$ using base 3:

$\log_2(3) = \frac{\log_3(3)}{\log_3(2)} = \frac{1}{\log_3(2)}$

Step 5: Final Expression

Substitute this into the expression:

$\frac{1}{2} \cdot \frac{1}{\log_3(2)} \cdot \log_3(4)$

Since $\log_3(4) = 2 \cdot \log_3(2)$, we can simplify:

$\frac{1}{2} \cdot \frac{1}{\log_3(2)} \cdot 2 \cdot \log_3(2)$

The $\log_3(2)$ terms cancel out, leaving:

$1$

Final Answer:

$\log_2(3) \cdot \log_9(4) = 1$

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Would you like more details on the steps or have any questions about the process?

Here are some related questions:

1. How can you simplify expressions involving multiple logarithms with different bases?
2. What is the change of base formula and when is it most useful?
3. Can logarithms with different bases be added or subtracted directly?
4. What are the properties of logarithms that help in simplification?
5. How do you solve equations that involve logarithms with different bases?

Tip: When working with logarithms, always consider using the change of base formula to convert them into a common base for easier manipulation.