We can simplify the expression using logarithmic properties. The relevant properties are:

1. Logarithm of a product: $\log_a(x) + \log_a(y) = \log_a(xy)$
2. Logarithm of a quotient: $\log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right)$

The given expression is:

$\log(400) + \log(20) - \log(8) + \log(100)$

Step 1: Combine the logarithms

First, combine the logarithms based on the properties:

$\log\left(\frac{400 \times 20 \times 100}{8}\right)$

Step 2: Simplify the inside expression

Now, calculate the value inside the logarithm:

$\frac{400 \times 20 \times 100}{8} = \frac{800000}{8} = 100000$

Step 3: Apply the logarithm

Finally, apply the logarithm:

$\log(100000)$

Since $\log(10^5) = 5$, we get:

$\log(100000) = 5$

Thus, the simplified value of the expression is 5.

Would you like more details, or have any questions?

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Here are 5 related questions to extend your understanding:

1. How do logarithmic properties help in simplifying expressions?
2. Can you simplify $\log(50) + \log(2) - \log(10)$?
3. What happens if we change the base of the logarithms, say to base 2?
4. How do we deal with logarithmic expressions involving square roots, like $\log(\sqrt{50})$?
5. What is the value of $\log(xy)$ in terms of $\log(x)$ and $\log(y)$?

Tip: Remember that logarithms convert multiplication into addition and division into subtraction, making them useful for simplifying complex products and quotients.