What is the significance of using a log scale in bode plots?

[influx]

How did they get the blue box? I know where they got the 0.5 inside the brackets from but not sure about the rest.

The bode plot in my data book has log10(wt) as the x-axis..

 

[NascentOxygen]

The formula in the blue box: that's how you convert from a voltage ratio into decibels, 20.log₁₀(...)

 

[NascentOxygen]

The bode plot in my data book has log10(wt) as the x-axis..

That's the frequency axis, they have normalized it and used a log scale so the graph can cover a wide frequency range without needing a 2 metre-wide sheet of graph paper. They could have used log-log graph paper instead of taking logs of the data values.