$\frac{d^n}{dx^n}[log(ax+b)]$ is equal to:

Let $y=\log { \left( ax+b \right) }$, let $D=\cfrac { d }{ dx }$

$\cfrac { dy }{ dx } =Dy=\cfrac { a }{ ax+b }$

${ D }^{ 2 }y=\cfrac { -{ a }^{ 2 } }{ { \left( ax+b \right) }^{ 2 } } ,{ D }^{ 3 }y=\cfrac { 2{ a }^{ 3 } }{ { \left( ax+b \right) }^{ 3 } }$

${ D }^{ 4 }y=\cfrac { -3.2.1.{ a }^{ 4 } }{ { \left( ax+b \right) }^{ 4 } }$

$\therefore { D }^{ n }y=\cfrac { { \left( -1 \right) }^{ n-1 }\left( n-1 \right) !{ a }^{ n } }{ { \left( ax+b \right) }^{ n } }$

$\therefore \cfrac { { d }^{ n } }{ d{ x }^{ n } } \left( \log { \left( ax+b \right) } \right) =\cfrac { { \left( -1 \right) }^{ n-1 }\left( n-1 \right) !{ a }^{ n } }{ { \left( ax+b \right) }^{ n } }$