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.