Below is a comparison of the monetary base and CPI since 1918. Each series has been indexed so that the value is 100 starting in 1918. The data is also presented in logarithmic scale so that it's easier to interpret rate of change. I'll admit there's not a lot of science involved in this analysis, but eyeballing the chart it's fairly clear that inflation and monetary base growth are highly correlated. Although the series diverge in the late depression/WWII era, they generally exhibit a similar pattern with a slightly lagged increase in CPI following an increase in the base.
If the two series are re-indexed to 100 in 1945, the relationship is even more clear. Between 1945 and 1983 CPI increases at almost the exact same pace as the monetary base. In January 1983, when the two series diverge, the CPI calculation was adjusted to substitute a survey of "owner's equivalent rent" for housing prices. Further adjustments were made under the Clinton administration which slow the pace of CPI growth.
Since 2008 we are seeing another divergence in the relationship between CPI and the monetary base, but as I've noted in other posts, the relationship of the monetary base to oil and gold remains extremely strong, which is consistent with the idea that when more money is printed it becomes less valuable. It is likely that a commensurate increase in CPI will occur at some point.