TBO \( 7^{\log _{\frac{1}{7}} \log _{\frac{1}{2}}(-x)}<2^{\log _{\frac{1}{2}} \log _{\frac{1}{2}}(-x)} \)

To solve the inequality \( 7^{\log _{\frac{1}{7}} \log _{\frac{1}{2}}(-x)} < 2^{\log _{\frac{1}{2}} \log _{\frac{1}{2}}(-x)} \), we can simplify both sides. First, we know that using the change of base formula, we can rewrite \( \log_{\frac{1}{7}} y \) as: \[ \log_{\frac{1}{7}} y = -\log_{7} y \] So, we have: \[ 7^{\log_{\frac{1}{7}} \log_{\frac{1}{2}} (-x)} = 7^{-\log_{7} \log_{\frac{1}{2}} (-x)} = \frac{1}{\log_{\frac{1}{2}} (-x)} \] For the right-hand side: \[ 2^{\log_{\frac{1}{2}} \log_{\frac{1}{2}}(-x)} = \frac{1}{\log_{\frac{1}{2}}(-x)} \] Putting it back into the inequality, we have: \[ \frac{1}{\log_{\frac{1}{2}} (-x)} < \frac{1}{\log_{\frac{1}{2}} (-x)} \] To ensure that the inequality holds, we can reason log properties. The left-hand side can also be represented with definitions of logarithms. The transformation is all in terms of negative logarithms thus giving us something like this: Next, we need to manipulate the logarithms and check for conditions under which it holds: 1. \( -x < 0 \) or \( x > 0 \) for the logs to be defined. 2. We know \( \log_{\frac{1}{2}}(-x) < \log_{7}(-x) \). At this point, we need to analyze it further mathematically. But if we rearranged last few steps we could say that if we rewrite them in terms of common bases, typically those inequalities would indicate values between some bases etc. To solve the initial statement would simply lead us back or canceled each other out, which implies constraints clearly define. Thus, we can further delve deeper \( LHS\) < \( RHS \) make any adjustments, keeping in mind all logarithm properties together. In this case, the inequality may not be possible to resolve solvable for specific x-values directly, but this manipulation offers the bounds under \(x\) which must stay positive throughout. More detailed computations further extend into direct numeric checking, resolving via numeric intervals depending on the inequalities simply conclude \(x > certain value\) required throughout. Thus providing meaningful exploration through these bases essentially pivots around logarithmic identities through succinct checking formation.