To solve the expression $5\log(4) \cdot 4\log(125)$, we can proceed as follows:

Step 1: Use the logarithm power rule

The power rule of logarithms states that $a\log(b) = \log(b^a)$. Applying this rule:

$5\log(4) = \log(4^5)$ $4\log(125) = \log(125^4)$

Step 2: Calculate the powers

Next, calculate the values:

$4^5 = 1024$ $125^4 = (5^3)^4 = 5^{12} = 244140625$

So, the expression becomes:

$\log(1024) \cdot \log(244140625)$

Step 3: Multiply the logarithms

Now, the expression $\log(1024) \cdot \log(244140625)$ is the final simplified form. If you need to approximate or find a decimal value, you would typically use a calculator or logarithm tables.

However, since the expression was not explicitly requesting a numerical answer, we can leave it as:

$\log(1024) \cdot \log(244140625)$

This is the simplified form.

Would you like more details or have any questions?

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Here are 8 questions to explore this topic further:

1. How would the solution change if different bases for the logarithms were used?
2. What is the significance of the logarithm power rule in simplifying expressions?
3. Can the properties of logarithms be used to expand or further simplify the given expression?
4. How does the choice of base affect the logarithmic value?
5. Can you calculate the numerical values of $\log(1024)$ and $\log(244140625)$ using a calculator?
6. How are logarithmic identities used in solving exponential equations?
7. What is the importance of understanding logarithms in calculus?
8. How can logarithms be applied in fields such as computer science and engineering?

Tip: When multiplying logarithmic expressions, remember that unless the logs are in the same base and are multiplied as arguments inside the logarithm, they cannot be directly combined.