II. Properties of L(x). In order to explore the properties of L(x) we will define several other functions. One important way to prove that two functions are equal is to first prove that they have the same derivative. Once you have shown that the derivatives are the same, you know that the functions only differ by a constant. By plugging in some known value for the functions you should be able to show that the constant is zero. (Throughout this section you may treat y as a constant.) 3) Define Ly(x)=dt. Prove Ly(x) = L(x). (Hint: find their derivatives.) 4) Prove L(xy) L(x) + L(y). (Hint: Use part (3) and properties of integrals) 5) Define f(x) = L(). Prove f(x) = -L(x). 6) Use (4) and (5) to prove that L (+) = L(x) - L(y)