From the Appearance tile or menu, within the General tab, you are encouraged to upload a separate logo for your mobile store.
The only difference between your desktop logo and your mobile logo is the suggested size dimensions. The different size dimensions will improve the mobile experience and appearance of your store from a mobile device.
It is extremely important to note that when establishing size dimensions for the desktop OR mobile logo that sizing is done in the logo's vector or .svg (Scalable Vector Format) format and THEN exported as a .jpg or .png.
Resizing a .png logo could cause distortion and the result will most likely not be to your satisfaction. If you are having issues with this, you need to resize your logo in .svg (vector) format and then export to the appropriate formats (.png or .jpg) for upload.
A few useful resources:
1) To convert a .png file to .svg go here:
2) To crop and export the .svg (vector) image, go here: