Setup:

Needs["Graphics`Graphics`"];
Needs["Graphics`ParametricPlot3D`"];
?CylindricalPlot3D

Problem 5:

Clear["Global`*"];

Part a) Log[z]

CylindricalPlot3D[
   Log[ r E^(I th) ] //Re
  ,{r,0,2}
  ,{th,-Pi ,+Pi}
  ,BoxRatios->{1,1,1}
  ];

CylindricalPlot3D[
   Log[ r E^(I th) ] //Im
  ,{r,0,2}
  ,{th,-Pi * 0.99,+Pi * 0.99}
  ,BoxRatios->{1,1,1}
  ];

CylindricalPlot3D[
   (Log[ r ] + I th) //Im
  ,{r,0.1,2}
  ,{th,-Pi ,+5 Pi }
  ,BoxRatios->{1,1,2}
  ,PlotPoints->31
  ];

((
  ( Log[ r E^( I Pi *a) ] 
  - Log[ r E^(-I Pi *a) ] 
  )  //Im  
     //PowerExpand 
 )   /.{a->1}
)

Plot[
{Log[ r E^(I Pi  *0.999) ] 
,Log[ r E^(-I Pi *0.999) ] } //Im //Evaluate
,{r,0.001,5}];

Problem 9:

Clear["Global`*"];

Part a)

Plot[ {E^(+I t), E^(-I t)} //Re //Evaluate
   ,{t,-4 Pi, 4 Pi}
   ,PlotStyle->{{Thickness[0.005]}
               ,{Thickness[0.015],Dashing[{0.06}]}
               }
  ];