By Bethany Bell
BBC News, Jerusalem
Israel is a country where prime ministers are often judged on their ability to deal with military crises.
Mr Olmert has so far been tough and uncompromising
The pressure on Ehud Olmert, who came to power just a few months ago, is now intense.
Unlike his predecessor, Ariel Sharon, Mr Olmert does not have significant military experience.
But so far he has taken a tough and uncompromising stance.
He has authorised the biggest strikes on Lebanon since Israel withdrew from the south of the country in 2000.
And he has emphatically rejected negotiations or a prisoner exchange with Hezbollah and Hamas in order to secure the release of the three captured Israeli soldiers.
He says Israel will not stop its attacks on Lebanon until the soldiers are released, until there is an end to militant rocket fire into Israel, and until a UN resolution on disarming Hezbollah is implemented.
For now the Israeli public and the coalition government appear to have rallied around Mr Olmert.
"I have moved to the right in recent days," one Israeli woman in Jerusalem told me. "Israel is being attacked."
Israelis are likely to back attacks as long as casualties are low
Alternatives to retaliation are rarely debated.
But opinion could change if the soldiers are not released soon or if casualties rise - either among civilians in Israel or among Israeli soldiers fighting in Lebanon and the Gaza Strip.
Mr Olmert's demands for the disarmament and disbanding of Hezbollah are unlikely to be met anytime soon.
Hezbollah has a unique and powerful role in Lebanon's political scene. Most observers believe it would be virtually impossible to disarm the group by force.
Some people here are concerned that Mr Olmert, in his desire to appear strong, has failed to consider an exit strategy if the current military strikes fail to achieve his objectives.
There are fears that the tactics could lead Israel into a broader conflict that could destabilise the entire region.
The crisis could also affect Mr Olmert's election platform to withdraw from parts of the West Bank.
Israel's Tourism Minister, Yitzhak Herzog, told the BBC that security concerns could affect plans to hand over more West Bank land to Palestinian control.
"Whilst Prime Minister Olmert and I... believe that we can pull out of most of the West Bank, we now have second thoughts about the viability of such a process if there are no new rules in the region."