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: 

```
{ComplexContourPlot[ReIm[z], {z, -3 - 3 I, 3 + 3 I}, PlotLabel -> z],
  ComplexContourPlot[ReIm[f[z]], {z, -3 - 3 I, 3 + 3 I}, 
   PlotLabel -> f[z]]} // Grid[{#}] &
```

The result:

 [![https://i.sstatic.net/7stvk.png][1]][1]


  [1]: https://i.sstatic.net/1QUer.png