When COVID-19 hit the United States, the numbers just seemed to explode. First, there were only one or two cases. Then there were 10. Then 100. Then thousands and then hundreds of thousands. Increases like this are hard to understand. But exponents and logarithms can help make sense of those dramatic increases.
Scientists often describe trends that increase very dramatically as being exponential. It means that things don’t increase (or decrease) at a steady pace or rate. It means the rate changes at some increasing pace.
An example is the decibel scale, which measures sound pressure level. It is one way to describe the strength of a sound wave. It’s not quite the same thing as loudness, in terms of human hearing, but it’s close. For every 10 decibel increase, the sound pressure increases 10 times. So a 20 decibel sound has not twice the sound pressure of 10 decibels, but 10 times that level. And the sound pressure level of a 50 decibel noise is 10,000 times greater than a 10-decibel whisper (because you’ve multiplied 10 x 10 x 10 x 10).
An exponent is a number that tells you how many times to multiply some base number by itself. In that example above, the base is 10. So using exponents, you could say that 50 decibels is 104 times as loud as 10 decibels. Exponents are shown as a superscript — a little number to the upper right of the base number. And that little 4 means you’re to multiply 10 times itself four times. Again, it’s 10 x 10 x 10 x 10 (or 10,000).
Logarithms are the inverse of exponents. A logarithm (or log) is the mathematical expression used to answer the question: How many times must one “base” number be multiplied by itself to get some other particular number?
For instance, how many times must a base of 10 be multiplied by itself to get 1,000? The answer is 3 (1,000 = 10 × 10 × 10). So the logarithm base 10 of 1,000 is 3. It’s written using a subscript (small number) to the lower right of the base number. So the statement would be log10(1,000) = 3.
At first, the idea of a logarithm might seem unfamiliar. But you probably already think logarithmically about numbers. You just don’t realize it.
Let’s think about how many digits a number has. The number 100 is 10 times as big as the number 10, but it only has one more digit. The number 1,000,000 is 100,000 times as big as 10, but it only has five more digits. The number of digits a number has grows logarithmically. And thinking about numbers also shows why logarithms can be useful for displaying data. Can you imagine if every time you wrote the number 1,000,000 you had to write down a million tally marks? You’d be there all week! But the “place value system” we use allows us to write down numbers in a much more efficient way.
Why describe things as logs and exponents?
Log scales can be useful because some types of human perception are logarithmic. In the case of sound, we perceive a conversation in a noisy room (60 dB) to be just a bit louder than a conversation in a quiet room (50 dB). Yet the sound pressure level of voices in the noisy room might be 10 times higher.
Another reason to use a log scale is that it allows scientists to show data easily. It would be hard to fit the 10 million lines on a sheet of graph paper that would be needed to plot the differences from a quiet whisper (30 decibels) to the sound of a jackhammer (100 decibels). But they’ll easily fit on a page using a scale that’s logarithmic. It’s also an easy way to see and understand big changes such as rates of growth (for a puppy, a tree or a country’s economy). Any time you see the phrase “order of magnitude,” you’re seeing a reference to a logarithm.
Logarithms have many uses in science. pH — the measure of how acidic or basic a solution is — is logarithmic. So is the Richter scale for measuring earthquake strength.
In 2020, the term logarithmic became best known to the public for its use in describing the spread of the new pandemic coronavirus (SARS-CoV-2). As long as each person who got infected spread the virus to no more than one other person, the size of the infection would stay the same or die out. But if the number was more than 1, it would increase “exponentially” — which means that a logarithmic scale could be useful to graph it.
Basic bases
The base number of a logarithm can be almost any number. But there are three bases which are especially common for science and other uses.
- Binary logarithm: This is a logarithm where the base number is two. Binary logarithms are the basis for the binary numeral system, which allows people to count using only the numbers zero and one. Binary logarithms are important in computer science. They’re also used in music theory. A binary logarithm describes the number of octaves between two musical notes.
- Natural logarithm: A so-called “natural” logarithm — written ln — is used in many areas of math and science. Here the base number is an irrational number referred to as e, or Euler’s number. (The mathematician Leonhard Euler did not intend to name it after himself. He was writing a math paper using letters to represent numbers and happened to use e for this number.) That e is about 2.72 (though you can never write it down completely in decimals). The number e has some very special mathematical properties that make it useful in many areas of math and science, including chemistry, economics (the study of wealth) and statistics. Researchers also have used the natural logarithm to define the curve that describes how a dog’s age relates to a human one.
- Common logarithm: This is a logarithm where the base number is 10. This is the logarithm used in measurements for sound, pH, electricity and light.
Power Words
More About Power Wordsacidic: An adjective for materials that contain acid. These materials often are capable of eating away at some minerals such as carbonate, or preventing their formation in the first place.
base: (in math) A number to be multiplied by itself in a logarithmic statement (and shown as a subscript to the lower right of the base number) or by the number of times called for by an exponent (shown as a superscript to the upper right of that base number). (in chemistry) A chemical that produces hydroxide ions (OH-) in a solution. Basic solutions are also referred to as alkaline. (in genetics) A shortened version of the term nucleobase. These bases are building blocks of DNA and RNA molecules.
binary: Something having two integral parts. (in mathematics and computer science) A number system where values are represented using two symbols 1 (on) or 0 (off).
chemistry: The field of science that deals with the composition, structure and properties of substances and how they interact. Scientists use this knowledge to study unfamiliar substances, to reproduce large quantities of useful substances or to design and create new and useful substances.
computer science: The scientific study of the principles and use of computers. Scientists who work in this field are known as computer scientists.
coronavirus: A family of viruses named for the crown-like spikes on their surface (corona means “crown” in Latin). Coronaviruses cause the common cold. The family also includes viruses that cause far more serious infections, including SARS.
COVID-19: A name given the coronavirus that caused a massive outbreak of potentially lethal disease, beginning in December 2019. Symptoms included pneumonia, fever, headaches and trouble breathing.
data: Facts and/or statistics collected together for analysis but not necessarily organized in a way that gives them meaning. For digital information (the type stored by computers), those data typically are numbers stored in a binary code, portrayed as strings of zeros and ones.
decibel: A measurement scale used for the intensity of sounds that can be picked up by the human ear. It starts at zero decibels (dB), a sound hardly audible to people with good hearing. A sound 10 times louder would be 10 dB. Because the scale is logarithmic, a sound 100 times louder than 0 dB would be 20 dB; one that’s 1,000 times louder than 0 dB would be described as 30 dB.
digit: (in math) An individual numeral (from 0 to 9) used to represent a number or some part of a number.
e: A mathematical constant that never changes. It is roughly, 2.718281828459. e stands for Euler’s number, the mathematician who discovered it. It’s the base number of a natural logarithm.
earthquake: A sudden and sometimes violent shaking of the ground, sometimes causing great destruction, as a result of movements within Earth’s crust or of volcanic action.
economics: The social science that deals with the production, distribution and consumption of goods and services and with the theory and management of economies or economic systems. A person who studies economics is an economist.
economy: Term for the combined wealth and resources (people, jobs, land, forests and minerals, for instance)of a nation or region. It is often measuredin terms of jobs and income or in terms of the production and use of goods (such as products) and services (for instance, nursing or internet access).
electricity: A flow of charge, usually from the movement of negatively charged particles, called electrons.
exponent: A number, shown as a superscript (tiny number to the upper right of some other “base” number or mathematical expression). An exponent identifies how many times that base number or expression must be multiplied by itself.
expression: (in mathematics) A statement that involves combinations of numbers and/or letters (that signify numbers that may vary) and includes directions (or rules) about what to do with those numbers (such as add or divide them, take their logarithm or make combinations of them equal one another).
infection: A disease that can spread from one organism to another.It’s usually caused by some type of germ.
inverse: Something that is the opposite or reverse of another thing, or that moves in the opposite direction to something.
irrational: (in math) A number that cannot be written as a whole number or fraction. When written as a decimal number, its digits never end or repeat. Examples are π (pi), the ratio of a circle’s diameter to its circumference (3.14159…), and e, Euler’s number (2.71828…).
log: (in math) An abbreviation for logarithm.
logarithm: The power (or exponent) to which one base number must be raised — multiplied by itself — to produce another number. For instance, in the base 10 system, 10 must be multiplied by 10 to produce 100. So the logarithm of 100, in a base 10 system, is 2. In base 10, the logarithm of 1,000 would be 3, the log of 10,000 would be 4, and so on.
magnitude: (in geology) A number used to describe the relative size of an earthquake. It runs from 1 to more than 8 and is calculated by the peak ground motion as recorded by seismographs. There are several magnitude scales. One of the more commonly used ones today is known as the moment magnitude. It’s based on the size of a fault (crack in Earth’s crust), how much the fault slips (moves) during a quake, and the energy force that was required to permit that movement. For each increase in magnitude, an earthquake produces 10 times more ground motion and releases about 32 times more energy. For perspective, a magnitude 8 quake can release energy equivalent to detonating 6 million tons of TNT.
numerical: Having to do with numbers.
octave: (in music) The interval between one note and the note with twice its frequency. There are actually 12 half steps in an octave, all the same size. Octaves are a pattern of sound differentiations typical of Northern and Western music.
pandemic: An epidemic that affects a large proportion of the population across a country or the world.
perception: The state of being aware of something — or the process of becoming aware of something — through use of the senses.
pH: A measure of a solution’s acidity or alkalinity. A pH of 7 is perfectly neutral. Acids have a pH lower than 7; the farther from 7, the stronger the acid. Alkaline solutions, called bases, have a pH higher than 7; again, the farther above 7, the stronger the base.
place value system: (in math) Numbers are expressed using 10 symbols — 0 to 9 — known as digits. When a number gets to be 10 or higher, no new symbols are used. Instead, we start a new column of digits to the left, which describes how many 10’s are in this number. After that, we write the digit indicating how many 1’s follow it. So the number ten is written 10 (for one 10 and zero 1’s). Twenty seven is written 27, because it has two 10’s and seven 1’s. Once a number exceeds 99, a new column is needed to identify the number of 100’s, followed by the number of 10’s and 1’s. And each time a number exceeds the spaces available, a new column is added, allowing us to show 1,000’s, 10,000’s, 100,000’s, millions and more.
radiation: (in physics) One of the three major ways that energy is transferred. (The other two are conduction and convection.) In radiation, electromagnetic waves carry energy from one place to another. Unlike conduction and convection, which need material to help transfer the energy, radiation can transfer energy across empty space.
SARS-CoV-2: A coronavirus that emerged in Wuhan, China, in late December 2019. It would go on to cause widespread — and sometimes lethal — disease throughout China and many other nations. Its name reflects its close similarity to the original coronavirus known as SARS (for severe acute respiratory syndrome). That SARS virus sparked a global outbreak of disease in 2003.
sound wave: A wave that transmits sound. Sound waves have alternating swaths of high and low pressure.
statistics: The practice or science of collecting and analyzing numerical data in large quantities and interpreting their meaning. Much of this work involves reducing errors that might be attributable to random variation. A professional who works in this field is called a statistician.
seismic wave: A wave traveling through the ground produced by an earthquake or some other means.
theory: (in science) A description of some aspect of the natural world based on extensive observations, tests and reason. A theory can also be a way of organizing a broad body of knowledge that applies in a broad range of circ*mstances to explain what will happen. Unlike the common definition of theory, a theory in science is not just a hunch. Ideas or conclusions that are based on a theory — and not yet on firm data or observations — are referred to as theoretical. Scientists who use mathematics and/or existing data to project what might happen in new situations are known as theorists.
virus: Tiny infectious particles consisting of RNA or DNA surrounded by protein. Viruses can reproduce only by injecting their genetic material into the cells of living creatures. Although scientists frequently refer to viruses as live or dead, in fact no virus is truly alive. It doesn’t eat like animals do, or make its own food the way plants do. It must hijack the cellular machinery of a living cell in order to survive.
wave: A disturbance or variation that travels through space and matter in a regular, oscillating fashion.
About Bethany Brookshire
Bethany Brookshire was a longtime staff writer atScience News Exploresand is the author of the bookPests: How Humans Create Animal Villains. She has a Ph.D. in physiology and pharmacology and likes to write about neuroscience, biology, climate and more. She thinks Porgs are an invasive species.
