The first thing is to understand what the function is doing. Like most other operators/functions, is the inverse of another, like square and square root.
Next, consider why we use dB. First it's as you mentioned, to make the values smaller and more manageable. Another reason to use dB is graphing the frequency response. Notice in your table of gain ratios versus dB, the gain ratio values are increasing exponentially, 1, 10, 100, 1000, ... Meanwhile, dB values are increasing linearly, 0, 20, 40, 60. This makes it easier to interpret and compare frequency plots. Distortion shows up as a curve. Easy to see if the graph should be straight in the dB plot. Hard to see in an already curving exponential plot.
As far as working out the dB value of the amplification, you've already written the formula:
Just plug in each measured/calculated amplification ratio into the formula, and create a table or graph of the results, similar to your example table. Relating the formula to log...
NOTE: When plotting/graphing log/dB data you need to use semi-log graph paper (log scale only on horizontal axis).