Permutations

To find the number of solutions for the equation that belong to the domain of definition of , we need to follow these steps:

1. Analyzing the domain of the logarithm:
   The expression inside the logarithm, , must be greater than zero for the logarithm to be defined. Thus, the domain of is:
   • and , which gives , or
   • and , which is not possible since both terms cannot be negative simultaneously.
2. Solving the quadratic equation:
   We solve by considering two cases, depending on the value of which affects :
   • For , , so the equation becomes: . Using the quadratic formula, , where , , : , , , .
   • For , , thus: . This equation is also solved using the quadratic formula, resulting in: , , .
     • (which lies outside our domain filter for ).

Conclusion: The number of solutions satisfying the original equation and falling within the domain of the logarithmic expression is 3.