Find all $\displaystyle k \in R$ so that inequalities $\displaystyle log_{(x-1)^2}(x+5)^4\geq1$ and $\displaystyle log_{x+k}(x+3)\leq1$ have no mutual solutions.
Can you show us what you have tried (Hint: try expanding the definitions using log laws for change of base and powers and get an expression with k on one side of the inequality)?