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The problem asks you to compare the values of \(5^3\) and \(10^2\) by expressing them in the same base with index 2.

### Step 1: Calculate the values of \(5^3\) and \(10^2\)
- \(5^3 = 5 \times 5 \times 5 = 125\)
- \(10^2 = 10 \times 10 = 100\)

### Step 2: Write them in the same base with index 2
We need to express both 125 and 100 as powers of 2:

- For \(5^3 = 125\):
  \[
  125 = 5^3 = \left( \sqrt{5} \right)^6 = \left( 2^{\log_2{5}} \right)^3
  \]
  Thus, \(5^3 = 2^{3\log_2{5}}\).

- For \(10^2 = 100\):
  \[
  100 = 10^2 = \left( 2^{\log_2{10}} \right)^2
  \]
  Thus, \(10^2 = 2^{2\log_2{10}}\).

Now let's compare these two expressions:
- \(5^3 = 2^{3\log_2{5}}\)
- \(10^2 = 2^{2\log_2{10}}\)

### Step 3: Compare the exponents
To compare the values, compare the exponents \(3\log_2{5}\) and \(2\log_2{10}\).

We know that:
- \(\log_2{5} \approx 2.32\)
- \(\log_2{10} \approx 3.32\)

Now, calculate:
\[
3\log_2{5} \approx 3 \times 2.32 = 6.96
\]
\[
2\log_2{10} \approx 2 \times 3.32 = 6.64
\]

Since \(6.96 > 6.64\), we conclude that \(5^3 > 10^2\).

Would you like more details or have any questions?

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### Related Questions:
1. How can you express \(5^4\) and \(10^3\) in the same base with index 2?
2. What is the value of \(\log_2{5}\) and \(\log_2{10}\) exactly?
3. How do you convert logarithms from one base to another?
4. What is the general method to compare any two numbers using logarithms?
5. How does the logarithmic comparison change if the base of the logarithm is different?

### Tip:
When comparing powers with different bases, expressing both in the same base using logarithms can help you make direct comparisons.

Compare the value of 5^3 and 10^2. Write them in the same base with index 2.

Learn how to compare 5^3 and 10^2 by expressing both numbers in the same base using logarithms. This problem involves key mathematical concepts such as exponentiation, logarithms, and base conversion. Suitable for students in Grades 9-12.

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