Project Nayuki

Extending the use of logarithmic scales

Introduction

Most quantities in this world are reported using linear units (e.g. 10 dollars, 3 metres per second, 5 kilograms, 2 people), and some using percentages (e.g. 60% of kids agree, price fell by 20%). Only occasionally are quantities reported on logarithmic scales (e.g. 90-decibel traffic noise, pH 4 acid).

As an intellectual exercise that’s not very practical, we can extend the usages of logarithmic scales in two ways:

  • Equate the various logarithmic units and use them outside their usual field of application. For example 10 dB ≅ 2.30 nepers, so saying “please lower the sound volume by 10 dB” is equivalent to saying “please lower the sound volume by 2.30 nepers”. This is similar in principle to linear units, where it’s normal to say “1 foot = 0.3048 metre”.

  • Report quantities using logarithmic units when they are normally not reported logarithmically. “I drove 1609 km” means the same as “I drove 32.07 dB(km)”. “The cost of gasoline is 75% of what it was yesterday” means the same as “The cost is −1.2 dB compared to yesterday”.

Existing units and usages

Here is an overview of fields where logarithmic scales are commonly used in practice. This establishes a frame of reference which we will later extend.

Engineering and music

10 decibels (dB) represents a 10× increase in power. So 1 decibel represents an increase of 100.1 ≅ 1.259×. Moreover, 10 decibels (dB) = 1 bel (B).

Decibels are used to measure sound intensity (e.g. the ambient noise on the street is 80 dB (SPL), i.e. 0.2 Pa), the power of electrical signals (e.g. this antenna is receiving radio signals at −65 dBm, i.e. 10−6.5 mW), and other electrical phenomena (cable attenuation per kilometre, voltage levels, etc.).

In music, an octave represents a 2× increase in frequency. There are 12 semitones per octave and 100 cents per semitone. So a semitone is a factor of 21/12 ≅ 1.0595, and a cent is a factor of 21/1200 ≅ 1.000578.

A decade is a factor of 10, used to describe frequencies. It’s used in electrical engineering / analog signal processing to describe the behavior of filters. A decade is equal to log2 10 ≅ 3.322 octaves. For example, a filter whose high-frequency roll-off is −10.0 dB per decade can also be described as being −3.01 dB per octave.

Science

The pH scale measures the concentration of hydrogen ions (H+ or H3O+) in a solution. pH 0 means 100 ions of H+ per litre of solution, while pH 14 means 10−14 H+ ions per litre. Notice how the scale is inverted compared to all the others; larger numbers mean less concentration.

The obsolete Richter scale measures the magnitude of earthquakes, where 1 unit means 10× the amplitude of vibrations. (This is not the same as power, though.)

Photography

Because the power of light entering a camera can easily vary by 6 orders of magnitude, thinking about the quantities logarithmically makes it easier to reason about them. An exposure value (EV) is a factor of 2. A stop or an f-stop represents the same factor. These three units differ by their connotations: EV is geared toward the final image brightness, f-stop is for describing aperture size, and stop is for any exposure variable that the photographer can control (shutter speed, aperture size, ISO sensitivity, flash brightness, image brightness).

Note that while ISO sensitivity is measured on a linear scale, on some video cameras it’s expressed instead as a signal gain measured logarithmically in decibels.

Information theory

Logarithmic units are used in information theory when we want to concisely quantify the number of possible outcomes a random variable has. For example, a roll of a single die has 6 outcomes, and this can be described using log2 6 ≅ 2.58 bits on average. Rolling two distinguishable dice has 36 outcomes, which takes log2 36 = 2 log2 6 ≅ 5.17 bits to describe. Twenty dice has 620 = 3 656 158 440 062 976 outcomes, which takes 51.7 bits.

A ban is the information content of a decimal digit, so 1 ban = log2 10 ≅ 3.322 bits. The nat is a unit based on the natural logarithm, thus 1 ban = ln 10 ≅ 2.302 nats. Don’t forget that 1 byte = 8 bits.

Pure math

When we look at ratios of various quantities, these ratios are dimensionless numbers. Normally percentages are used (e.g. this road is 30% longer than the other), but we can also use units like nepers (a factor of e, approximately 2.718), centinepers (1 neper = 100 centinepers), log points (synonym of centineper), and orders of magnitude (a factor of 10) to describe these ratios. For example, +30% = 1.30× ≅ +0.262 nepers ≅ +26.2 centinepers (or log points) ≅ +0.114 order of magnitude.

Extended non-standard uses

Now let’s start to apply logarithmic scales and units in ways that sound weird with respect to the conventional usages:

  • Make your photo brighter by 3 dB (i.e. 1.995×).

  • Please quiet down by 2 stops (i.e. 1/4×).

  • The commodity price gained 5 log points yesterday but fell 4 log points today (i.e. +1.005%).

  • A millionaire owns at least 60 dB(USD); a billionaire has 90 dB(USD).

  • My cousin’s height was 165 cm and grew by 12.96 millibels this year (i.e. +3.03% to 170 cm).

  • The lab apparatus reached a cold −90 dB(K) (i.e. 1 nanokelvin).

  • The SD card has a capacity of 38.5 gigabans (i.e. 16 gigabytes).

Note that the dB unit is already used quite flexibly with a variety of absolute reference points. Namely, 10 dB(thing) means 10× of thing, where thing could be micrometres, kilonewtons, dollars, or such.

The big equality of units is what makes this possible. All of the following quantities are equal by definition:

  • 1 / (8 ln 2) ≅ 0.1803369 byte or octet
  • 1 / (ln 10) ≅ 0.4342945 bel or decade or ban
  • 1 / (ln e) = 1.000000 nat or neper
  • 1 / (ln 2) ≅ 1.442695 bit or EV or f-stop or octave
  • 10 / (ln 10) ≅ 4.342945 decibel or deciban
  • 100 / (ln e) = 100.0000 log point or centineper

(For example, 1 nat = 0.1803369 byte, and 1.442695 EV = 100 log points.)

Applications in finance

A sequence of percentage changes can be misleading. For example, if a stock price rises by 10% then drops by 10%, the overall change is actually −1%, not zero. If instead we use logarithmic measures, there is no such problem: A rise of 1 dB followed by a fall of 1 dB is exactly equal to 0 dB, no change.

Here’s a plausible example of this deceptive arithmetic: A mutual fund’s 5-year history of annual returns is −70%, +30%, +50%, +20%, +40%. Naively, the total return for the 5-year period looks like +70%. But in fact it’s −1.72%, because the reciprocal function makes the −70% very “heavyweight” compared to the positive changes. Using log points instead, the 5-year sequence is approximately −120.4, +26.2, +40.5, +18.2, +33.6. The sum of these rounded numbers is −1.9 (nowhere near +60.0), which agrees well with the correct result of about −1.735 log points (i.e. −1.72%).

Similarly, compound interest can be approximated by simple interest for small numbers, but not for big numbers. For example, 5% annual interest compounded for 4 periods is approximately 20%, but is actually 21.550625%. By contrast, 5 log points of annual interest (about 5.127%) compounded for 4 periods is necessarily equal to 20 log points overall (about 22.140%).

As a bonus, log points are approximately equal to percentages for small quantities, thanks to the exponential function having a derivative of 1 at the origin. Namely, if we say a price increases (or decreases) by n log points, then it’s almost the same as saying that the price increases (or decreases) by n percent.

If you’re feeling adventurous, you can play with measuring financial investment returns in log points instead of in percentages. Who knows where this new perspective might take you?

(Someone else did write some thoughts about this concept. A Loonie Saved (blog): Log points, Dogbert and log points.)