You came up with:

f(x) = -Inx + Cx + D

=> f(1) = -ln(1) + C + D = 0

=> C + D = 0 ...................(1) since, ln(1) = 0

=> f(2) = -ln(2) + 2C + D = 0

=> 2C + D = ln(2).............(2)

So now to find C and D, we must solve the system:

C + D = 0 ...................(1)

2C + D = ln(2) .............(2)

=> C = ln(2) ...........(2) - (1)

but C + D = 0

=> ln(2) + D = 0

=> D = -ln(2)

so f(x) = -lnx + ln(2)x - ln(2)