Problem: Verify properties of natural log. One way to define the natural log is lnx =...

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Problem: Verify properties of natural log. One way to define the natural log is lnx = This definition is sufficient for most properties that we are used to: (1) In(1) 0 (2) In (#) = Inx - Iny (3) In (+) = -Inx = (4) In(xy) Inx+lny (5) In() rlnz Begin by verifying property 1 using properties of definite integrals. The key to verifying properties 2,4,5 is to start with the quantity on the left and express it as an integral. Followed by making the appropriate substitution to handle the bounds. Property 3 follows from properties 1 and 2. Hint: for property 5 it is easier to write t = f(u) and dt = f'(u) du instead of the other way around. Warning: In order to receive credit you must verify the properties using the integral definition. Problem: Verify properties of natural log. One way to define the natural log is lnx = This definition is sufficient for most properties that we are used to: (1) In(1) 0 (2) In (#) = Inx - Iny (3) In (+) = -Inx = (4) In(xy) Inx+lny (5) In() rlnz Begin by verifying property 1 using properties of definite integrals. The key to verifying properties 2,4,5 is to start with the quantity on the left and express it as an integral. Followed by making the appropriate substitution to handle the bounds. Property 3 follows from properties 1 and 2. Hint: for property 5 it is easier to write t = f(u) and dt = f'(u) du instead of the other way around. Warning: In order to receive credit you must verify the properties using the integral definition.