I now look forward to applying the analytical, problem-solving nature I have cultivated since elementary school to solve the major problems faced by society.
Does this need to be replaced by to solving
I now look forward to applying the analytical, problem-solving nature I have cultivated since elementary school to solve the major problems faced by society.
Does this need to be replaced by to solving
Your original sentence is correct.
to applying something to solve something
I now look forward to applying the analytical, problem-solving nature I have cultivated since elementary school to solve the major problems faced by society.
Alternatively, you might say
to use something for something
I now look forward to using the analytical, problem-solving nature I have cultivated since elementary school for solving the major problems faced by society
or possibly you want
to apply something to something
I now look forward to applying the analytical, problem-solving nature I have cultivated since elementary school to the major problems faced by society.
"solving" the ills of society is implicit in the last sentence.
to applying something to solving something
is awkward in its expression.