The decibel and Mr. Dow

The decibel scale is the logarithmic scale ubiquitously employed in plotting the magnitude of frequency responses. Unfortunately, students and practioners often do not know the multiplicative factors that correspond to particular decibel values, making estimates of magnitudes nonintuitive. This paper derives “by hand“ rational approximations to the factors of the integer decibel values between 1 and 20 that are within 1.5% of the actual values. The approximations have denominators of 2, 4, 5, 8 or 10. The approximations are easy to remember and easy to re-derive without the availability of a calculator. Rational decibel approximations of the integer factors between 1 and 10 are also derived. These approximations are handy not only for estimating magnitudes, but also for back-of-the-envelope computations. An example of determining average compound growth rate of the Dow Jones industrial average from inception to 2006 without a calculator is provided for giving students practice using these approximations.