3. \( 4^{-x-2}=12 \)

To solve the equation \( 4^{-x-2}=12 \), we can start by rewriting the left side using the property of exponents. Recall that \( 4 \) can be expressed as \( 2^2 \), so we have: \[ (2^2)^{-x-2} = 12 \] This simplifies to: \[ 2^{-2(x+2)} = 12 \] Next, we recognize that \( 12 \) can be expressed as \( 2^2 \cdot 3 \), which allows for easier comparison of exponents: \[ 2^{-2x-4} = 2^2 \cdot 3 \] At this point, we can take the logarithm of both sides or manipulate the equation further. For simplicity, let's solve for \( x \) directly using logarithms: Taking the logarithm base 2 (since we have base 2 on the left) gives us: \[ -2x - 4 = \log_2(12) \] Now isolate \( x \): \[ -2x = \log_2(12) + 4 \] \[ x = -\frac{\log_2(12) + 4}{2} \] Using the change of base formula, we can find \( \log_2(12) \). Therefore, we have the value of \( x \) represented in terms of logarithms.