To solve \( \log_{16} 64 \), we need to express the logarithm in a solvable form.

We want to find \( x \) such that:

\[
16^x = 64
\]

First, express both 16 and 64 as powers of 2:

- \( 16 = 2^4 \)
- \( 64 = 2^6 \)

Now substitute these values into the equation:

\[
(2^4)^x = 2^6
\]

Simplify:

\[
2^{4x} = 2^6
\]

Since the bases are the same, equate the exponents:

\[
4x = 6
\]

Solve for \( x \):

\[
x = \frac{6}{4} = \frac{3}{2}
\]

Thus, \( \log_{16} 64 = \frac{3}{2} \).

Would you like further details or have any questions?

Here are five related questions to expand on this concept:
1. What is \( \log_{4} 64 \)?
2. Can you solve \( \log_{64} 16 \)?
3. How would you calculate \( \log_{32} 16 \)?
4. What is the value of \( \log_{81} 9 \)?
5. How would you solve \( \log_{16} 256 \)?

**Tip**: When working with logarithms, converting numbers to powers of a common base (e.g., powers of 2) often simplifies the problem.

This guide explains how to solve the logarithmic equation log16 64 by converting the numbers into powers of 2. Learn step-by-step how to apply logarithmic and exponentiation properties to find that log16 64 equals 3/2. Suitable for high school students in Grades 9-12.

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