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.