How would you evaluate \left(\log2\right)\left(\log50\right)+\left(\log5\right)\left(\log20\right)-\left(\log100\right)\left(\log5\right)\left(\log2\right) without a calculator?

To evaluate the expression \(\left(\log 2\right)\left(\log 50\right)+\left(\log 5\right)\left(\log 20\right)-\left(\log 100\right)\left(\log 5\right)\left(\log 2\right)\) without a calculator, we can use the properties of logarithms and some simplifications.

### Step 1: Express all logarithms in terms of \(\log 2\) and \(\log 5\).
Given:
\[
\log 50 = \log (2 \times 5^2) = \log 2 + 2\log 5
\]
\[
\log 20 = \log (2^2 \times 5) = 2\log 2 + \log 5
\]
\[
\log 100 = \log (10^2) = 2\log 10 = 2(\log 2 + \log 5)
\]

### Step 2: Substitute into the original expression.
Substitute the above logarithm values into the original expression:

\[
\left(\log 2\right)\left(\log 50\right) = \log 2 \times (\log 2 + 2\log 5)
\]
\[
\left(\log 5\right)\left(\log 20\right) = \log 5 \times (2\log 2 + \log 5)
\]
\[
\left(\log 100\right)\left(\log 5\right)\left(\log 2\right) = [2(\log 2 + \log 5)] \times \log 5 \times \log 2
\]

The original expression becomes:

\[
(\log 2)(\log 2 + 2\log 5) + (\log 5)(2\log 2 + \log 5) - 2(\log 2 + \log 5)(\log 5)(\log 2)
\]

### Step 3: Distribute and simplify.
Expand each term:

1. \(\log 2 \times (\log 2 + 2\log 5) = (\log 2)^2 + 2\log 2 \cdot \log 5\)
2. \(\log 5 \times (2\log 2 + \log 5) = 2\log 5 \cdot \log 2 + (\log 5)^2\)
3. The last term simplifies as:
   \[
   2(\log 2 + \log 5) \cdot \log 5 \cdot \log 2 = 2(\log 2 \cdot \log 5 + (\log 5)^2 \cdot \log 2)
   \]

Substituting these into the full expression:

\[
(\log 2)^2 + 2\log 2 \cdot \log 5 + 2\log 2 \cdot \log 5 + (\log 5)^2 - [2\log 2 \cdot \log 5 + 2(\log 5)^2 \cdot \log 2]
\]

### Step 4: Combine like terms.
Combine the terms, noticing that certain terms will cancel out:

\[
(\log 2)^2 + (\log 5)^2 + 4\log 2 \cdot \log 5 - 2(\log 2 \cdot \log 5 + \log 2 \cdot \log 5) = (\log 2)^2 + (\log 5)^2
\]

### Final Result:
Thus, the expression simplifies to:

\[
\boxed{(\log 2)^2 + (\log 5)^2}
\]

This is the simplified form of the given expression.

Would you like more details, or do you have any questions?

### Related Questions:
1. How would you evaluate \((\log 3)(\log 50) + (\log 5)(\log 30) - (\log 150)(\log 2)(\log 5)\) without a calculator?
2. What are the properties of logarithms used to simplify expressions like this?
3. How can you evaluate \((\log 2 + \log 3)(\log 6) - \log 18 \cdot \log 3\) without a calculator?
4. Can this expression be evaluated differently using change of base formula?
5. What would be the result if \(\log 2\) and \(\log 5\) were known numerical values?

### Tip:
Remember that logarithms can often be expressed in terms of simpler bases, which can simplify complex logarithmic expressions.

Learn how to evaluate the logarithmic expression (log2)(log50) + (log5)(log20) - (log100)(log5)(log2) step-by-step without a calculator. This problem involves properties of logarithms and simplification techniques. Suitable for advanced high school students and those preparing for logarithmic expression problems.

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