\( F ( x ) < \log _ { \frac { 1 } { 2 } } ( 4 x + 4 ) - 2 \)

To analyze the inequality \( F(x) < \log_{\frac{1}{2}}(4x + 4) - 2 \), we can start by simplifying the right-hand side. Recall that the logarithm with base \(\frac{1}{2}\) can be rewritten using the change of base formula: \[ \log_{\frac{1}{2}}(y) = -\log_2(y) \] Thus, we can rewrite the expression: \[ \log_{\frac{1}{2}}(4x + 4) = -\log_2(4x + 4) \] Now substituting this back into the inequality gives: \[ F(x) < -\log_2(4x + 4) - 2 \] Next, we can express \(-2\) in terms of logarithms: \[ -2 = -\log_2(4) = \log_2\left(\frac{1}{4}\right) \] So, we can combine the logarithmic terms: \[ F(x) < -\log_2(4x + 4) - \log_2(4) = -\log_2(4(4x + 4)) = -\log_2(16x + 16) \] This leads us to: \[ F(x) < -\log_2(16(x + 1)) \] Now, we can express this in terms of positive logarithms: \[ F(x) < \log_2\left(\frac{1}{16(x + 1)}\right) \] This gives us a clearer form of the inequality. The final result is: \[ F(x) < \log_2\left(\frac{1}{16(x + 1)}\right) \] This inequality can be used to analyze the function \(F(x)\) in relation to the logarithmic expression. If you have specific values or further context for \(F(x)\), we can explore that as well.