Short answer: an electronic logbook is legal; the FAA will accept almost anything as an endorsement; it's often most practical to collect endorsements on paper but electronic versions are also fine.
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For commercial motor vehicles operating in the United States, electronic logbooks are now mandated by law. Making the right decision when selecting this technology can deliver significant dividends in efficiency and productivity.
An ELD can be on a smartphone or other wireless device if the device meets the ELD rule's technical specifications. If the device is a portable it must be mounted in a fixed position during commercial motor vehicle (CMV) operation and visible to the driver from a normal seated driving position.