To evaluate $\log_{64}(32)$, we need to find the power to which 64 must be raised to get 32.

Solution:

1. Express 64 and 32 in terms of powers of 2:

   • $64 = 2^6$
   • $32 = 2^5$

2. Rewrite $\log_{64}(32)$ using these powers of 2: $\log_{64}(32) = \log_{2^6}(2^5)$

3. Use the change of base formula: $\log_{2^6}(2^5) = \frac{5}{6}$

Conclusion:

$\log_{64}(32) = \frac{5}{6}$

Would you like further details or have any questions?

Related Questions:

1. How would you solve $\log_{64}(16)$?
2. What is $\log_{8}(32)$?
3. Can you express $\log_{64}(1)$ and explain why?
4. How can we apply the change of base formula in general?
5. What is the value of $\log_{125}(25)$?

Tip: When working with logarithms, converting bases to powers of the same number often simplifies the calculation.