Question and given condition

The question states: “The value of log_n 4^-16 is -32. Find the value of n.”

In mathematical form, this is written as:
log_n(4^-16) = -32.

Step-by-step detailed solution

1. Convert logarithmic form to exponential form

By definition of a logarithm, if log_a b = c, then a^c = b.
Applying this to log_n(4^-16) = -32 gives:
n^-32 = 4^-16.

2. Remove the negative exponents

Rewrite both sides with positive exponents:
n^-32 = 4^-16 ⇒ 1 / n³² = 1 / 4¹⁶.
Since the fractions are equal, the denominators must be equal, so
n³² = 4¹⁶.

3. Express 4 as a power

Note that 4 can be written as 2², and hence
4¹⁶ = (2²)¹⁶ = 2³².
Therefore, we have n³² = 4¹⁶.

4. Match exponents to find n

Observe that 4¹⁶ is the same as (4¹)¹⁶, and we already have
n³² on the left side. To compare exponents conveniently, take both sides to the
power 1⁄16:
(n³²)^1/16 = (4¹⁶)^1/16, which yields
n² = 4.

Solving n² = 4 gives n = ±2, but the base of a logarithm must be positive and not equal to 1.
Therefore, the valid value is n = 2.

5. Verification of the solution

Substitute n = 2 back into the original equation:
log₂(4^-16) =
log₂((2²)^-16) =
log₂(2^-32) = -32.
This matches the given value, confirming that n = 2 is correct.

Final answer

The base n that satisfies log_n(4^-16) = -32 is:

n = 2