To simplify \(\frac{\log(125)}{\log(\sqrt{5})}\), let's break it down step by step:

1. **Express the numbers in terms of powers of 5**:
   - \( 125 = 5^3 \)
   - \( \sqrt{5} = 5^{1/2} \)

2. **Apply logarithmic properties**:
   Using the power rule of logarithms, \(\log(a^b) = b \log(a)\), we can simplify both the numerator and the denominator.

   - \( \log(125) = \log(5^3) = 3 \log(5) \)
   - \( \log(\sqrt{5}) = \log(5^{1/2}) = \frac{1}{2} \log(5) \)

3. **Substitute these values back into the original expression**:
   \[
   \frac{\log(125)}{\log(\sqrt{5})} = \frac{3 \log(5)}{\frac{1}{2} \log(5)}
   \]

4. **Simplify**:
   Cancel out \(\log(5)\) from the numerator and the denominator:
   \[
   \frac{3 \log(5)}{\frac{1}{2} \log(5)} = \frac{3}{\frac{1}{2}} = 3 \times 2 = 6
   \]

Thus, the simplified value is:

\[
\boxed{6}
\]

Would you like further explanation, or do you have any questions about this process?

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Here are five related questions:
1. How would you simplify \(\frac{\log(1000)}{\log(10)}\)?
2. Can you explain how to use the change of base formula for logarithms?
3. What is \(\log_b(a^n)\) in terms of simpler logarithmic expressions?
4. How does the logarithmic property \(\log(a \cdot b) = \log(a) + \log(b)\) apply in calculations?
5. What is the value of \(\log(1)\) and why?

**Tip**: Remember that logarithmic rules apply to any base, not just base 10, as long as the base is consistent.

Discover how to simplify the logarithmic expression log(125) / log(sqrt(5)) step by step, using properties of exponents and logarithms. Suitable for students in Grades 9-12, this guide covers key logarithmic concepts and properties.

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