You could use ```ComplexContourPlot``` for visualising the real and imaginary axes on the complex ```z``` plane under the complex exponential mapping ```Exp[z]```.

Define function for complex mapping: 

```f[z_] := Exp[z]```

Then use ```ComplexContourPlot``` for visualising the contours: 

Edit: ```Contours``` and ```ContourLabels``` added to ```ComplexContourPlot``` following [Michael E2][1] comment. This shows more clearly that the same lines are displayed in both contour plots.

```
{ComplexContourPlot[ReIm[z], {z, -3 - 3 I, 3 + 3 I}, PlotLabel -> z, 
   Contours -> {Range[-2, 2]}, ContourLabels -> All],
  ComplexContourPlot[ReIm[f[z]], {z, -3 - 3 I, 3 + 3 I}, 
   PlotLabel -> f[z], Contours -> {Range[-2, 2]}, 
   ContourLabels -> All]} // Grid[{#}, Frame -> True] &
```

The result:

[![enter image description here][2]][2]


Also you could look into the modulus and argument of ```Exp[z]```. Perhaps this shows more clearly the mapping:

```
{ComplexContourPlot[AbsArg[z], {z, -3 - 3 I, 3 + 3 I}, PlotLabel -> z,
    Contours -> {Range[-3, 3]}, ContourLabels -> All],
  ComplexContourPlot[AbsArg[f[z]], {z, -3 - 3 I, 3 + 3 I}, 
   PlotLabel -> f[z], Contours -> {Range[-3, 3]}, 
   ContourLabels -> All]} // Grid[{#}, Frame -> True] &
```

[![enter image description here][3]][3]


  [1]: https://mathematica.stackexchange.com/users/4999/michael-e2
  [2]: https://i.sstatic.net/99eXK.png
  [3]: https://i.sstatic.net/916ML.png


Here you can see that circles in ```z``` plane (```Abs[z]```constant) are mapped into real lines ```Exp[x+Iy] (*  Exp[x] is constant *)```, and that lines through the origin in the ```z``` plane (```Arg[z]```constant) are mapped into ```Exp[x+Iy] (* y is constant*)``` lines.