Mr Cameron said the British Government and everyone in Britain stood with Japan and would do all we could to help the country recover.
The PM said the courage, resilience and dedication shown by the people of Japan was a “true inspiration”.
Mr Cameron wrote in the book:
This comes with the deepest sympathy and condolences from the British Government and British people to everyone in Japan who has suffered from the earthquake and tsunami.
We are your friend and stand with you at this time.
We will do all we can to help you recover as we know you will.
Indeed the courage and resilience of everyone in Japan, including the nuclear workers and emergency services has been inspirational and the true spirit of your great country.
Read more: Britain doing all it can to support those affected by the Japanese earthquake