To solve $2 \log \left( \frac{1}{125} \right)$, let's break it down step by step.

Step 1: Use the logarithmic property

One of the logarithmic properties says that:

$a \log(x) = \log(x^a)$

So, applying this property:

$2 \log \left( \frac{1}{125} \right) = \log \left( \left( \frac{1}{125} \right)^2 \right)$

Step 2: Simplify the expression inside the logarithm

Next, simplify the expression inside the logarithm:

$\left( \frac{1}{125} \right)^2 = \frac{1^2}{125^2} = \frac{1}{15625}$

So the expression becomes:

$\log \left( \frac{1}{15625} \right)$

Step 3: Use the logarithmic property for division

The logarithmic property for division states that:

$\log \left( \frac{1}{x} \right) = -\log(x)$

Thus, applying this property:

$\log \left( \frac{1}{15625} \right) = -\log(15625)$

Step 4: Calculate $\log(15625)$

If the logarithm is base 10 (common logarithm), then:

$\log(15625) \approx 4.1938$

Thus, the expression becomes:

$- \log(15625) \approx -4.1938$

Final Answer:

$2 \log \left( \frac{1}{125} \right) \approx -4.1938$

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

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Here are five related questions:

1. What is the value of $\log \left( \frac{1}{10} \right)$?
2. Can you simplify $\log \left( \frac{1}{1000} \right)$?
3. What is the result of $3 \log \left( \frac{1}{100} \right)$?
4. How do you evaluate $\log(100000)$?
5. What is the logarithmic value of a negative power of 10?

Tip: Remember that logarithms of fractions can be transformed using the negative log rule $\log \left( \frac{1}{x} \right) = -\log(x)$.