L.A. County’s unemployment rate held steady at 4.5 percent in August for the third straight month, even as the county recorded sizable job gains, according to state figures released Friday. The steady unemployment rate reflected little movement in the county’s 5.15 million-strong labor force in August. According to the state Employment Development Department figures, the level of unemployment extrapolated from a monthly survey of households in the county showed a decline of 2,000 to 233,000, while employment dropped by 1,000 to 4.92 million and the labor force increased by 1,000 to 5.15 million. The unemployment rate was below the 4.6 percent L.A. County recorded in August of last year. But it was higher than the 4.2 percent unemployment rate the state registered last month and the 3.9 percent national figure. County employers added 23,500 jobs to their payrolls in August to reach 4,476,700, reversing the trend of substantial declines over the previous two months due to seasonal job losses in the education sector. The health care and social services sector was the biggest job gainer in August, up 7,900 jobs. Among the other gainers: motion pictures and sound recording, up 5,500 jobs; and retail trade, up 2,200.