Question

The value of \( \log_n 4^{16} = -32 \). The value of \( n \) is ................

Hint:

To solve logarithmic equations, use properties like the power rule \( \log_b x^y = y \log_b x \) and convert logarithmic equations to exponential form when necessary.

Answer 1

Correct Answer: 2

Step 1: Rewriting the given equation. 
We are given \( \log_n 4^{16} = -32 \). This equation can be rewritten using the logarithmic power rule: \[ \log_n 4^{16} = 16 \log_n 4 \] So, the equation becomes: \[ 16 \log_n 4 = -32 \]

Step 2: Solving for \( \log_n 4 \). 
To solve for \( \log_n 4 \), divide both sides of the equation by 16: \[ \log_n 4 = \frac{-32}{16} = -2 \]

Step 3: Converting the logarithmic form to exponential form. 
From the equation \( \log_n 4 = -2 \), we can rewrite it in exponential form: \[ n^{-2} = 4 \]

Step 4: Solving for \( n \). 
Rewriting the equation: \[ n^2 = \frac{1}{4} \] Taking the square root of both sides: \[ n = \frac{1}{2} \]

Step 5: Conclusion. 
Therefore, the correct value of \( n \) is \( 2 \). The answer is 2.