Replace a lost car log book in Ireland

The short answer
In Ireland, the document most people call the car "log book" is the Vehicle Registration Certificate, usually shortened to VRC. If yours is lost, the registered owner can apply for a replacement through the official motor tax and vehicle registration service, either online or through a local Motor Tax Office. A fee applies. The replacement is issued to the owner and address already recorded for the vehicle, so those details need to be correct before you apply.
One qualification changes the answer for some readers. If you are not the registered owner, for example because you bought the car and the change of ownership was never completed, you generally cannot request a duplicate in your own name. In that case the ownership record has to be sorted out first, normally with the help of the previous owner or the dealer involved.
What the log book is and what it is not
The VRC records the vehicle's registration number, its identifying and technical details, and the registered owner. It is the document used when transferring ownership on a sale. It shows who is registered as responsible for the vehicle, which is not always the same as legal title. A car on finance, for instance, may legally belong to the finance company while the VRC lists the person who drives it.
Irish readers may also come across UK guidance for the V5C, which is the British equivalent. The two documents serve a similar purpose, but the UK replacement process and fees do not apply to Irish-registered vehicles.
Steps to take before applying
Search for the original first. A duplicate is meant to replace the original, and an old VRC that turns up later should not be used. Check that the owner name and address on record are current. If you have moved, update the address so the replacement is not posted to your old home. If the document was stolen, or was taken with the car or other papers, consider reporting it to An Garda Síochána. A stolen VRC could be misused in a fraudulent sale. Have your vehicle registration number and suitable identification ready, and be ready to pay the replacement fee.
Details to confirm with the official service
The source material for this guide did not include current figures or procedural specifics. Before you apply, confirm the following with the official motor tax service or your local Motor Tax Office:
The current fee for a replacement VRC. Whether your vehicle and circumstances qualify for the online route or need an in-person or postal application. Which form and identification documents are currently required. How long the replacement usually takes to arrive. Any extra steps for company-registered, imported or recently transferred vehicles.
If you are selling or buying soon
A missing log book can hold up a private sale, because the change of ownership normally relies on it. If you are selling, apply for the replacement before advertising the car. If you are buying, treat a seller who cannot produce the VRC with caution. Ask why it is missing and wait for the replacement rather than relying on assurances that it will follow.
Bottom line
A lost car log book in Ireland is replaced by applying to the official vehicle registration and motor tax service for a duplicate Vehicle Registration Certificate. Only the registered owner can do this, and the replacement goes to the address on record. Check the current fee, accepted application routes and required documents with the official service before applying, because those details were not available for this guide.