7. If \( f(x)=\log 2^{x^{2}+4 x-5} \) then, what is the domain of \( \mathrm{f}(\mathrm{x}) \) ?

To find the domain of the function \( f(x) = \log 2^{x^{2} + 4x - 5} \), we need to ensure that the argument of the logarithm, which is \( 2^{x^{2} + 4x - 5} \), is positive. Since \( 2^{y} > 0 \) for all real \( y \), we only need the exponent \( x^{2} + 4x - 5 \) to be greater than 0. We can solve \( x^{2} + 4x - 5 > 0 \) by first finding its roots. Factoring gives us \( (x + 5)(x - 1) > 0 \). The critical points from this factorization are \( x = -5 \) and \( x = 1 \). Now, we analyze the intervals divided by these critical points: 1. For \( x < -5 \): Choose \( x = -6 \); \( (-6 + 5)(-6 - 1) = (-1)(-7) > 0\). 2. For \( -5 < x < 1 \): Choose \( x = 0 \); \( (0 + 5)(0 - 1) = (5)(-1) < 0\). 3. For \( x > 1 \): Choose \( x = 2 \); \( (2 + 5)(2 - 1) = (7)(1) > 0\). The domain of \( f(x) \) is the union of the intervals where the expression is positive: \( (-\infty, -5) \cup (1, \infty) \).