(Reuters) - The euro zone private sector grew at its fastest pace in six months in January as firms slashed prices at the steepest rate in nearly five years, a business survey showed on Wednesday.

Data compiler Markit said the survey, carried out mostly before the European Central Bank announced a near-trillion euro quantitative easing program to revive inflation and boost the economy, pointed to first-quarter growth of 0.3 percent.

That matches the median forecast in a Reuters poll last month and, if realized, would be faster than the 0.1 percent economists have penciled in for the end of 2014.

Markit's final January Composite Purchasing Managers' Index (PMI), seen as a good indicator of growth, stood at 52.6, higher than a preliminary estimate of 52.2 and December's 51.4.

"The euro zone enjoyed a positive start to 2015, as growth of economic activity accelerated. Among the big-four nations, output expanded in Germany, Italy and Spain, but the downturn in the French economy extended into its ninth month," Markit said.

However, that growth came at a cost to margins. An index measuring output prices fell to 46.9 from December's 48.1, its lowest reading since February 2010, suggesting firms were slashing prices to drum up trade.

Prices dropped a record-equaling 0.6 percent last month as commodity prices, Brent crude in particular, tumbled.

Price-cutting helped drive service industry activity up at the fastest rate in five months. The services sector PMI rose to 52.7 from December's 51.6, ahead of the flash 52.3 estimate.

Confidence about the ECB's QE program and signs of growth in new orders accelerating helped lift a gauge of optimism among service firms by the biggest one-month margin in over five years, to 63.9 from 59.7.