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

Learn how to simplify the expression 2 log(1/125) using properties of logarithms, including exponent rules and negative logarithms. This detailed step-by-step solution is suitable for students in Grades 8-10 and covers essential concepts in logarithms and exponents.

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