We can simplify the expression \( 2\log 50 + 2\log 8 - 2\log 100 \) using logarithmic properties. Here's the step-by-step solution:

### Step 1: Apply the logarithm power rule
The logarithm power rule states that \( a\log b = \log b^a \). Let's apply this to each term:

\[
2\log 50 = \log 50^2
\]
\[
2\log 8 = \log 8^2
\]
\[
2\log 100 = \log 100^2
\]

So the expression becomes:

\[
\log 50^2 + \log 8^2 - \log 100^2
\]

### Step 2: Simplify the exponents
Now calculate the squares:

\[
\log 50^2 = \log 2500
\]
\[
\log 8^2 = \log 64
\]
\[
\log 100^2 = \log 10000
\]

Thus, the expression is now:

\[
\log 2500 + \log 64 - \log 10000
\]

### Step 3: Apply the logarithm addition and subtraction rules
The logarithm addition rule states \( \log a + \log b = \log (a \times b) \), and the subtraction rule states \( \log a - \log b = \log \left(\frac{a}{b}\right) \).

First, combine the terms using the addition rule:

\[
\log (2500 \times 64) - \log 10000
\]

### Step 4: Perform the multiplication
Calculate \( 2500 \times 64 \):

\[
2500 \times 64 = 160000
\]

So the expression becomes:

\[
\log 160000 - \log 10000
\]

### Step 5: Apply the subtraction rule
Finally, apply the subtraction rule:

\[
\log \left(\frac{160000}{10000}\right)
\]

### Step 6: Simplify the division
Simplify the fraction:

\[
\frac{160000}{10000} = 16
\]

So the final simplified expression is:

\[
\log 16
\]

### Final Answer:
The expression simplifies to \( \log 16 \).

Would you like more details on any part of this process?

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**Relative Questions:**
1. How do you use the change of base formula for logarithms?
2. Can you explain the logarithm product rule in detail?
3. How do logarithm properties apply to solving logarithmic equations?
4. What are the common logarithms and their typical applications?
5. How would you expand or condense more complex logarithmic expressions?

**Tip:** Always remember that logarithms turn multiplication into addition and division into subtraction, which is handy when simplifying expressions.

Learn how to simplify the logarithmic expression 2log 50 + 2log 8 - 2log 100 step by step. This problem covers key logarithmic properties, including the power rule, addition rule, and subtraction rule. Suitable for students in Grades 9-11.

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