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.

Explore the step-by-step solution to simplifying the logarithmic expression log400 + log20 - log8 + log100 using the product and quotient properties of logarithms. Learn how to apply logarithmic rules and solve complex expressions. Suitable for Grades 10-12 students.

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