Hello, this is more like a tutorial for the beginners than a project who don't know how to work with logo construction grids in the Adobe Illustrator. Here I am trying to explain how to work with logo construction grid and how to make logo presentation showing construction guides step by step like the image given below.
Step 1 : First thing first, for every logo work you should have a good design in your mind with a great concept and  then create at least 10-15 sketches trying different variations and choose the best one to work with. Here I am sharing one of my work in which I tried to incorporate letter P with the elephant icon. As I am not so good in sketching so I am sharing just only one sketching which I did just to make this tutorial
Step 2 :  Now import your sketch into illustrator and name this layer to "Rough Sketch" and then lock this layer.
Step 3 : Create a new layer on the top of it and name this layer to "Logo Guides." Before doing anything first select in which grid size you want to work. I usually work with 8x8 grid size. You can change your grid size by going to Edit > Preferences > Guides & Grid and you can change grid size by changing subdivisions. Grid size depends on the complexity of your design. Complex design will need more subdivisions.
Step 4 : Turn on the Snap to grid option by going to View > Snap to grid and Show grid option by going to View > Show grid.
Now you work space will look like this.
Step 5 : Select the "Logo Guide" layer and with 1pt stroke (stroke thickness depends on you) without any fill color draw the outline of the logo by snipping to grid. Proper contact and cut is very important during this process as it will help in the next step building shape using Shape Builder Tool. After completing your design will look like this.
Step 6 : After creating outline turn of the Show grid and Snap to grid option and then create a new layer on the top of it and name that layer "Logo File." Copy the outline of logo on the new layer you created. You can do that with Ctrl + A (Selection)  > Ctrl + C (Copy) > Ctrl + F (Paste) I repeat use Ctrl + F to copy not Ctrl + V because Ctrl + F paste the file exactly just over it without any mismatch unlike Ctrl + V which paste randomly on the work space. After pasting the design over "Logo File" layer turn off the "Logo Guide" layer. It will look like this.
Step 7 : Now select the design with Ctrl + A and then select the Shape Builder Tool from the tool bar. Choose a fill color with no stroke and start filling only the portion which will be part of your final logo by dragging the arrow over it (It will show + sign with arrow) and to delete the unwanted portion press Alt button while dragging over it (It will show minus sign with arrow). I have shown in the image below: 
After completing this step you will have your design. You can experiment with the color later.
Step 8 : Now drag "Logo File" layer below the "Logo Guide" layer. You can make extra copy also of this layer to experiment with design.
Step 9 : Now turn on the "Logo Guide" Layer. Your design will look like this.You can change the style, color, opacity, thickness of the construction guide of your choice.
In case if you want that dotted logo guides then you can change the stroke into dashed line from the Stroke panel and save the file into your desired JPEG or PNG format. 
Now you are ready to share your design with the rest of the world showing your logo construction guides.
Thank you for watching this. Directly mail me at cullenx.art@gmail.com for any work and if you just want to have a chat with me join me on different social media.