What Are Logarithms?
Most scientific calculators only calculate logarithms in base 10 and base e.
Credit: Bildagentur Zoonar GmbH | Shutterstock


A logarithm is a mathematical operation that determines how many times a certain number, called the base, is multiplied by itself to reach another number. Because logarithms relate geometric progressions to arithmetic progressions, examples are found throughout nature and art, such as the spacing of guitar frets, mineral hardness, and the intensities of sounds, stars, windstorms, earthquakes and acids. Logarithms even describe how humans instinctively think about numbers.

Logarithms were invented in the 17th century as a calculation tool by Scottish mathematician John Napier (1550 to 1617), who coined the term from the Greek words for ratio (logos) and number (arithmos). Before the invention of mechanical (and later electronic) calculators, logarithms were extremely important for simplifying computations found in astronomy, navigation, surveying, and later engineering.

Logarithms characterize how many times you need to fold a sheet of paper to get 64 layers. Every time you fold the paper in half, the number of layers doubles. Mathematically speaking, 2 (the base) multiplied by itself a certain number of times is 64. How many multiplications are necessary? This question is written as:

log2(64) = x

A logarithm can be thought of as the inverse of an exponential, so the above equation has the same meaning as:

2x = 64

Since 2 x 2 x 2 x 2 x 2 x 2 = 64, 26 = 64. This means if we fold a piece of paper in half six times, it will have 64 layers. Consequently, the base-2 logarithm of 64 is 6, so log2(64) = 6.

When you take 1 milliliter of a liquid, add 99 ml of water, mix the solution, and then take a 1-ml sample, 99 out of every 100 molecules from the original liquid is replaced by water molecules, meaning only 1/100 of the molecules from the original liquid are left. Sometimes this is referred to as a “C dilution” from Roman numeral for a hundred. Understanding that 1 ml of pure alcohol has roughly 1022 (a one followed by 22 zeroes) molecules, how many C dilutions will it take until all but one molecule is replaced by water? Mathematically speaking, 1/100 (the base) multiplied by itself a certain number of times is 1/1022, so how many multiplications are necessary? This question is written as:

log1/100(1/1022) = 11

Thus, after 11 C dilutions, there will only be one molecule of the original alcohol left. (Aside, this is less than half of the 30 C dilutions common in homeopathy, which shows why the practice is irreconcilable with modern chemistry.)

Most scientific calculators only calculate logarithms in base 10, written as log(x) for common logarithm and base e, written as ln(x) for natural logarithm (the reason why the letters l and n are backwards is lost to history). The number e, which equals about 2.71828, is an irrational number (like pi) with a non-repeating string of decimals stretching to infinity. Arising naturally out of the development of logarithms and calculus, it is known both as Napier’s Constant and Euler’s Number, after Leonhard Euler (1707 to 1783), a Swiss mathematician who advanced the topic a century later.

To do a logarithm in a base other than 10 or e, we employ a property intrinsic to logarithms. From our first example above, log2(64) may be entered into a calculator as “log(64)/log(2)” or “ln(64)/ln(2)”; either will give the desired answer of 6. Likewise, log1/100(1/1022) equals “log(1/1022)/log(1/100)” and “ln(1/1022)/ln(1/100)” for an answer of 11.

Because logarithms relate multiplicative changes to incremental changes, logarithmic scales pop up in a surprising number of scientific and everyday phenomena. Take sound intensity for example: To increase a speaker’s volume by 10 decibels (dB), it is necessary to supply it with 10 times the power. Likewise, +20 dB requires 100 times the power and +30 dB requires 1,000 times. Decibels are said to “progress arithmetically” or “vary on a logarithmic scale” because they change proportionally with the logarithm of some other measurement; in this case the power of the sound wave, which “progresses geometrically” or “varies on a linear scale.”

 

Linear Scale

 

Logarithmic Scale

Sound intensity

Power [×10]

Decibels (dB) [+10]

Note pitch

Frequency [×2]

Note [+12 half steps]

Star brightness

Power per unit area [×100]

Magnitude [-5]

Earthquake intensity

Energy [×1000]

Richter Scale [+2]

Wind intensity

Wind speed [×1.5]

Beaufort Scale [+1]

Mineral hardness

Absolute hardness [×3 (approx.)]

Mohs Scale [+1]

Acidity/Basicity

Concentration of H+ions [×10]

pH [-1]

The table shows that the numbers relating various linear and logarithmic systems vary widely. This is because a logarithmic scale is often invented first as a characterization technique without a deep understanding of the measurable phenomena behind that characterization. A good example is star brightness, which was introduced by Hipparchus, a second-century B.C. Greek astronomer. The brightest stars in the night sky were said to be of first magnitude (m = 1), whereas the faintest were of sixth magnitude (m = 6). In the 19th century A.D., English astronomer Norman Robert Pogson discovered that magnitude is the logarithm of the amount of starlight that hits a detector.

Most other logarithmic scales have a similar story. That logarithmic scales often come first suggests that they are, in a sense, intuitive. This not only has to do with our perception, but also how we instinctively think about numbers.

Though logarithmic scales are troublesome to many (if not most) math students, they strangely have a lot to do with how we all instinctively thought about numbers as infants. Stanislas Dehaene, a professor at the Collège de France and an expert on numeral cognition, recorded the brain activity in two- to three-month-old infants to see how they perceive changes on a computer screen. A change from eight ducks to 16 ducks caused activity in the parietal lobe, showing that newborns have an intuition of numbers. An infant’s response is smaller the closer the numbers are together, but what’s interesting is how an infant perceives “closeness.” For example, eight and nine are perceived much closer to each other than one and two. According to Dehaene, “they seem to care about the logarithm of the number.” Basically, infants don’t think about differences, they think about ratios.

Research with people native to the Amazon, who “do not have number words beyond five, and they don’t recite these numbers,” shows that people, if left to their instincts, will continue thinking this way. If someone is shown one object on the left and nine on the right and is asked, “What is in the middle?”, you and I would choose five objects, but the average Amazonian will choose three. When thinking in terms of ratios and logarithmic scales (rather than differences and linear scales), one times three is three, and three times three is nine, so three is in the middle of one and nine.

John Napier’s 1614 work, "Mirifici Logarithmorum Canonis Descriptio" (Description of the Wonderful Canon of Logarithms), contained 90 pages of numerical tables relating to logarithms. These were of particular utility for simplifying calculations. In the following example, a method using logarithms takes advantage of the fact that it’s easier to add rather than multiply. The following example isn’t really made any simpler, but it does demonstrate the process of using logarithmic tables.

37 × 59

From a version of Napier’s tables, each of these numbers could be written as follows:

101.5682 ×  101.7709

Exponents have a useful property that enables the following step:

101.5682 + 1.7709

Which leaves:

103.3391

From another table, the final answer is determined:

2,183

This property of making multiplication analogous to addition enables yet another antiquated calculation technique: the slide rule. Two normal (linear) rulers can be used to add numbers as shown:

Linear rulers can be used to do addition. Here it is shown that 2 + 3 = 5.
Linear rulers can be used to do addition. Here it is shown that 2 + 3 = 5.
Credit: Robert J. Coolman

Similar to the procedure shown above, two rulers can be used to multiply when printed with logarithmic scales.

Logarithmic rulers can be used to do multiplication. Here it is shown that 2 × 8 = 16.
Logarithmic rulers can be used to do multiplication. Here it is shown that 2 × 8 = 16.
Credit: Robert J. Coolman

These markings also match the spacing of frets on the fingerboard of a guitar or ukulele. Musical notes vary on a logarithmic scale because progressively higher octaves (ends of a musical scale) are perceived by the human ear as evenly spaced even though they’re produced by repeatedly cutting the string in half (multiplying by ½). Between the neck and the mid-point of a guitar string, there will be 12 logarithmically spaced frets.

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