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To solve \( \log_{32} 4 \) without using technology, we’ll apply the change of base and properties of logarithms.

1. **Rewrite \( \log_{32} 4 \) in terms of exponentials**:
   The equation \( \log_{32} 4 = x \) implies that:
   \[
   32^x = 4
   \]

2. **Express 32 and 4 as powers of 2**:
   - \( 32 = 2^5 \)
   - \( 4 = 2^2 \)

   Substitute these into the equation:
   \[
   (2^5)^x = 2^2
   \]

3. **Simplify the equation**:
   Using the power rule \((a^m)^n = a^{m \cdot n}\), we get:
   \[
   2^{5x} = 2^2
   \]

4. **Set the exponents equal to each other**:
   Since the bases are the same, we can equate the exponents:
   \[
   5x = 2
   \]

5. **Solve for \( x \)**:
   \[
   x = \frac{2}{5}
   \]

Thus,
\[
\log_{32} 4 = \frac{2}{5}
\]

Would you like further details on any step, or have any questions? 

Here are five related questions to consider:

1. How would you solve \( \log_{32} 16 \) without using technology?
2. Can you evaluate \( \log_{4} 16 \) and compare the method?
3. How does the change of base formula apply to \( \log_{32} 4 \)?
4. What are the advantages of expressing logarithms in terms of smaller bases?
5. How do properties of logarithms simplify complex expressions?

**Tip**: Remember that expressing numbers as powers of a common base often makes logarithmic equations much easier to solve.

Evaluate \( \log_{32} 4 \) without using technology.

Learn how to evaluate \( \log_{32} 4 \) without using technology. This solution explains how to express both numbers as powers of 2 and apply the power rule of exponents, suitable for high school students in Grades 10-12.

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