Mr Prendergast, who was a senior officer with the Australian Federal Police, was named for the position in mid-July and arrived in the country this week.
The position had been vacant since the beginning of the year when his predecessor was forced to quit amid controversy over his age.
Mr Prendergast's appointment has drawn criticism from various local groups and members of the public but parliamentarians and government officials, including a former Prime Minister Manasseh Sogavare, welcome the choice.
His appointment comes as the country prepares for elections later this year.
The last Australian to serve as Solomon Islands Police Commissioner, Shane Castles, was dismissed in 2007 and barred from returning to Honiara as the government labelled him a threat to the country's public security and morality.