Logarithms

Definition

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting. [12]

The logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

Logarithmic scales condense broad quantities into narrower ranges. They support some models in psychophysics, aid in forensic accounting, appear in formulas counting prime numbers or roughly approximating factorials, and help to describe frequency ratios of musical intervals. [16]

A logarithm is a special type of math related to exponents. We don’t always know what the exponent will be and often want to find out. Logarithms, otherwise just known as “log,” simply states how many times we need to multiply a number by itself to get another number.

A log has a base, an exponent and an argument.

We often use log with a base ten because it helps us quickly understand very large numbers.

When we use logarithm base 10, we’re asking the question “how many times (the exponent) do I need to multiply 10 by itself to get a certain number (the argument)?” [13]

The base b logarithm of a number is the exponent that we need to raise the base in order to get the number. When b is raised to the power of y is equal x:

b^y = x

Then the base b logarithm of x is equal to y:

log_b(x) = y

The logarithmic scale⁴ can describe very big or very small numbers with shorter notation. [3]

Who

When I was in school (17+ years), logarithms were taught but not to the degree needed for use in real life. In some of the many positions I have held over the years, logarithms and logarithmic graphing were essential to the tasks I worked on. Use this page as a springboard in your understanding of logarithms and primarily log₁₀.

What

This page introduces you to logarithms and uses decibels and the Richter Scale to show you how they are used. Also, the section on Graphing Logarithmic Functions presents a starting point in understanding logarithmic graphs.

Why

Logarithms very useful in the field of science, technology, and mathematics. Here are a few examples of real-life applications of logarithms. [10]

Electronic calculators have logarithms to make our calculations much easier.
Logarithms are used in surveys and celestial navigation.
Logarithms can be used to calculate the level of noise in decibels.
Ratio active decay, acidity [PH] of a substance and Richter scale are all measured in logarithmic form.

See Theoretical Knowledge Vs Practical Application.

Decibels

Sound is all around us and can be measured to inform and protect us, as some sounds are not safe. In fact, loud noise can be very damaging to hearing. The level of noise, where a person is in relation to the noise (distance to the noise), and the amount of time they listen to it can all result in risk for hearing loss. Sound is measured in units called decibels (dB). The higher the decibel level, the louder the noise.¹

Decibel is a logarithmic unit that indicates ratio or gain and is used to indicate the level of acoustic waves and electronic signals. The dB level can be viewed as relative gain of one level vs. other level, or absolute logarithmic scale level for well known reference levels. Decibel is a dimensionless unit. The ratio in bels is the base 10 logarithm of the ratio of P₁ and P₀: [2]

Ratio_B = log₁₀ ( P₁ / P₀ )

Decibel is one tenth of a bel, so 1 bel is equal to 10 decibel: 1B = 10dB

Where:⁵

B – bels
Ratio_B is the sound intensity decibels
P₁ is the sound intensity (W/m^2)
P₀ is the reference sound intensity (equal to 10^-12 W/m^2 for ambient noise)

Richter Scale

The Richter scale, officially called the “Richter Magnitude Scale,” is a numerical value used to measure the power of earthquakes. It is a logarithmic scale based on the amplitude of waves recorded by a seismograph. This means that each whole number increase on the scale corresponds to an absolute increase by a factor of ten. Earthquakes measured at less than about 2.0 on the Richter scale are not very serious, and can barely even be measured, much less felt. An earthquake is usually considered much more serious, and is felt by most people, once it hits about 5.0.

It is theoretically possible to have an earthquake of 10.0 or stronger, although this has never been recorded. Such a quake would be classified as Massive, and cause devastation across a very wide area. [6]

Representation of Richter scale earthquake magnitudes and energy equivalents. Source: Bravo and Ortiz (2005); Andrea et al. (2011) – ResearchGate

The Richter magnitude of an earthquake is determined from the logarithm of the amplitude of waves recorded by seismographs. Adjustments are included for the variation in the distance between the various seismographs and the epicenter of the earthquakes. Moment Magnitude (MW) is based on physical properties of the earthquake derived from an analysis of all the waveforms recorded from the shaking. [7]

Logarithmic View of the Universe

From the scale of planet Earth, at a few thousands of kilometers, to the scale of the observable Universe, at nearly 100 billion light-years, there’s a long way from here to the cosmic horizon. But rather than a linear scale, which would take several quintillions of Earths lined end-to-end to reach the limits of the observable Universe, a logarithmic scale holds far more cosmic insights to an onlooker. [11]

In the Solar System, we typically measure distances in Astronomical Units (AU), where the Earth-Sun distance is 1 AU. Mercury and Mars are also about ~1 AU from Earth, with Saturn at ~10 AU, the Kuiper belt ending before ~100 AU, and the Oort cloud largely existing at ~10,000 AU. It’s an enormous distance on a linear scale, but only a small set of “factors of 10” away on a logarithmic one. – BIG THINK

Graphing Logarithmic Functions

Most people are familiar with reading numbers on a number line or reading data from a graph. However, under certain circumstances, a standard scale may not be useful. If the data grows or decreases exponentially, then you will need to use what is called a logarithmic scale. For example, a graph of the number of McDonald’s hamburgers sold over time would start at 1 million in 1955; then 5 million just a year later; then 400 million, 1 billion (in less than 10 years), and up to 80 billion by 1990.This data would be too much for a standard graph, but it is easily displayed on a logarithmic scale. You need to understand that a logarithmic scale has a different system of displaying the numbers, which are not evenly spaced as on a standard scale. By knowing how to read a logarithmic scale you can more effectively read and represent data in graphic form. [8]

Why are they common in science?

Plotting on a log-log scale has many advantages for environmental science.

You are often dealing with very big or very small numbers, and particularly you are often dealing with numbers that range over many orders of magnitude.

For instance, if you are measuring concentrations of some pollutant, and your data are something like 0.01 ppb, 0.2 ppb, 2ppb, 117 ppb, if you plot it on a linear scale, then the three smaller data points will all be “squished” in at almost the origin while the one big point sticks way out. You won’t be able to see your data at all. This is also why, for example, we use pH or the Richter scale, both of which use logarithms to compress the range of possible numbers.
Power laws are very common.

If you plot any power law (y = a * x^b) on a log-log scale, you get a straight line whose slope is b. So it’s always good to plot power law relationships on log-log axes, so you can see how closely they fit to a straight line.

On standard axes, power law and exponential relations often look very similar. On log-log axes, the power law looks very much like a straight line while the exponential relation does not (to make the exponential linear, you’ll have to take the log of only the y axis). [9]

How

An arithmetic progression is a sequence of numbers like 0, 1, 2, 3, 4, 5 ,6

A geometric progression is a sequence of numbers like 1, 3, 9, 27, 81, 243, 729

Logarithms [15] constitute a connection between an arithmetic and a geometric progression. Let’s see how pairing these two progressions solves the problem of multiply large numbers (versus addition), dividing large numbers (versus subtraction) and dividing by n versus finding the nth root.

For example, using our arithmetic and geometric progressions above. we create a table.

Geometric Progression	Arithmetic Progression
1	0
3	1
9	2
27	3
81	4
243	5
729	6

Example Table For Explaining Logarithms

Say we want to multiply two numbers from the geometric column: 3 * 81

We look at the corresponding numbers in the arithmetic column in this case they are 1 and 4. We add these numbers together: 1 + 4 = 5.

Then we look back at the corresponding number in the geometric column and we get 243. 243 is the result of 3 x 81. In this way, we transformed multiplication into addition.

The power of this method is that we can multiply more than two numbers together at the same time, the same applies to division but with division we subtract instead of adding, and the same goes for the nth root but with finding the nth root we divide by n. [14]

Many of the References and Additional Reading websites and Videos will assist you with logarithms.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

[1] “What Are Decibels (Db)? | Loudness Levels, Safe, Unsafe”. 2021. earq.com. https://www.earq.com/hearing-health/decibels.

[2] ⭐ “What Is A Decibel (Db)?”. 2021. rapidtables.com. https://www.rapidtables.com/electric/decibel.html.

[3] ⭐ “Log Rules | Logarithm Rules”. 2021. rapidtables.com. https://www.rapidtables.com/math/algebra/Logarithm.html.

[4] “Logarithmic Scale – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Logarithmic_scale.

[5] “Decibel Calculator – Calculator Academy”. 2021. Calculator Academy. https://calculator.academy/decibel-calculator/.

[6] “What Is The Richter Scale? (With Pictures)”. 2011. All Things Nature. https://www.allthingsnature.org/what-is-the-richter-scale.htm.

[7] Shepherd, Marshall. 2021. “Hurricane And Earthquake Scales Often Confuse People – Why That’s Dangerous”. Forbes. https://www.forbes.com/sites/marshallshepherd/2019/07/06/hurricane-and-earthquake-scales-often-confuse-people-why-thats-dangerous/?sh=1b4754e339ff.

[8] “How To Read A Logarithmic Scale: 10 Steps (With Pictures)”. 2021. Wikihow.Com. https://www.wikihow.com/Read-a-Logarithmic-Scale.

[9] Peterson, Dave. 2021. “Logarithmic Graphing – The Math Doctors”. themathdoctors.org. https://www.themathdoctors.org/logarithmic-graphing/.

[10] “Introduction to Logarithms – Explanation & Examples”. 2021. The Story of Mathematics. https://www.storyofmathematics.com/logarithm.

[11] “This Logarithmic View Of The Universe Will Blow Your Mind”. 2022. Big Think. https://bigthink.com/starts-with-a-bang/logarithmic-view-universe/.

[12] “Logarithm – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Logarithm.

[13] Data Scientist Dad. “What is a Logarithm?”. 2023. Medium. https://datascientistdad.medium.com/what-is-a-logarithm-107284de3471.

[14] ⭐ Merriam, Areeba. “The Invention Of Logarithms”. 2023. Medium. https://www.cantorsparadise.com/the-invention-of-logarithms-25294dcda7e8.

[15] Logarithm is derived from two Greek words Logos Airthmos which translates to proportional number. [14]

[16] DoubtConnect. “Logarithms Made Easy For JEE“. 2022. Medium. https://doubtconnect.medium.com/logarithms-made-easy-for-jee-4bf8ab370413.

Additional Reading

Decibels

“A Tutorial on the Decibel”. 2021. arrl.org. http://www.arrl.org/files/file/A%20Tutorial%20on%20the%20Decibel%20-%20Version%202_1%20-%20Formatted.pdf.

“Db: What Is A Decibel?”. 2021. animations.physics.unsw.edu.au. https://animations.physics.unsw.edu.au/jw/dB.htm.

“Decibel”. 2021. phys.hawaii.edu. https://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/decibel.pdf.

“Decibels Dba | Computer Fan Noise And Decibels”. 2021. silentpc.com. https://silentpc.com/articles/decibels.

“Decibel – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Decibel.

⭐ “Decibel Db – Formula & Definition »Electronics Notes”. 2021. electronics-notes.com. https://www.electronics-notes.com/articles/basic_concepts/decibel/basics-tutorial-formula-equation.php.

“Decibel Formula “. 2021. softschools.com. https://softschools.com/formulas/physics/decibel_formula/320/.

McNulty, Keith. 2025. “Possibly The Most Useful Beast In The History of All Mathematics.” Medium. Medium. July 2. https://blog.keithmcnulty.org/possibly-the-most-useful-beast-in-the-history-of-all-mathematics-af6053895568.

The logarithm was an invention which unlocked incredible untapped computational potential.

“Physics Tutorials – Sound – Decibel Levels”. 2021. internet4classrooms.com. https://www.internet4classrooms.com/sound_decibel.htm.

“How To Make Noise Calculations With Decibels – Womack Machine Supply Company”. 2021. Womack Machine Supply Company. https://www.womackmachine.com/engineering-toolbox/data-sheets/how-to-make-noise-calculations-with-decibels/.

Graphing Logarithmic Functions

“13.2: Logarithmic Functions And Their Graphs”. 2020. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Tradler_and_Carley)/13%3A_Exponential_and_Logarithmic_Functions/13.02%3A_Logarithmic_functions_and_their_graphs.

“Graphing Logarithmic Functions”. 2021. varsitytutors.com. https://www.varsitytutors.com/hotmath/hotmath_help/topics/graphing-logarithmic-functions.

“Graphing With Logarithmic Paper Tutorial | Physics”. 2021. physics.uoguelph.ca. https://www.physics.uoguelph.ca/graphing-logarithmic-paper-tutorial.

“Graphs of logarithmic functions (Algebra 2 level)”. 2021. Khan Academy. https://www.khanacademy.org/math/algebra-home/alg-exp-and-log#alg-graphs-of-logarithmic-functions.

“Graphs Of Logarithmic Function – Explanation & Examples”. 2021. storyofmathematics.com. https://www.storyofmathematics.com/graphs-of-logarithmic-functions.

Logarithms

” Algebra – Logarithm Functions “. 2021. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/Classes/Alg/LogFunctions.aspx.

“Basic idea and rules for logarithms – Math Insight”. 2023. mathinsight.org. https://mathinsight.org/logarithm_basics.

A logarithm is the opposite of a power. In other words, if we take a logarithm of a number, we undo an exponentiation.

⭐ “Basic Log Rules & Expanding Log Expressions”. 2023. purplemath.com. https://www.purplemath.com/modules/logrules.htm.

You have learned various rules for manipulating and simplifying expressions with exponents, such as the rule that says that x³ × x⁵ equals x⁸ because you can add the exponents. There are similar rules for logarithms.

Log Rules:

1) log_b(mn) = log_b(m) + log_b(n)
2) log_b(m/n) = log_b(m) – log_b(n)
3) log_b(mⁿ) = n · log_b(m)

In less formal terms, the log rules might be expressed as:

1) Multiplication inside the log can be turned into addition outside the log, and vice versa.
2) Division inside the log can be turned into subtraction outside the log, and vice versa.
3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.

Warning: Just as when you’re dealing with exponents, the above rules work only if the bases are the same. For instance, the expression “log_d(m) + log_b(n)” cannot be simplified, because the bases (the “d” and the “b”) are not the same, just as x² × y³ cannot be simplified because the bases (the x and y) are not the same.

“Exponentials and Logarithms Cheat Sheet”. 2021. pmt.physicsandmathstutor.com. https://pmt.physicsandmathstutor.com/download/Maths/A-level/Pure/Exponentials-and-Logarithms-1/Cheat-Sheets/Exponentials%20and%20Logarithms.pdf.

“Introduction To Logarithms”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/logarithms.html.

“Logarithm Cheat Sheet – DoubleRoot.In”. 2021. doubleroot.in. https://doubleroot.in/cheat-sheets/logarithm/.

“Logarithms | Algebra 2 | Math | Khan Academy”. 2021. Khan Academy. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs#x2ec2f6f830c9fb89:log-intro.

“Logarithms | Brilliant Math & Science Wiki”. 2021. brilliant.org. https://brilliant.org/wiki/logarithms/.

“Log Rules | Logarithm Rules”. 2021. rapidtables.com. https://www.rapidtables.com/math/algebra/Logarithm.html.

“Log Rules – Natural Log Rules (Rules of ln) | Logarithm Rules”. 2023. CUEMATH. https://www.cuemath.com/algebra/log-rules/.

Log rules refer to the rules of logarithms. These rules are derived from the rules of exponents as a logarithm is just the other way of writing an exponent. The logarithm rules are used:

1. to compress a group of logarithms into a single logarithm
2. to expand a logarithm into a group of logarithms

Let us learn more about log rules and we will solve some example problems using the logarithm rules.

⭐ “Logarithms: The Long Forgotten Story Of Scientific Progress”. 2022. Medium. https://medium.com/math-simplified/logarithms-the-long-forgotten-story-of-scientific-progress-11a4412ccb4b.

“Log Table: Logarithm Table With Examples & Questions- Embibe”. 2021. Embibe Exams. https://www.embibe.com/exams/log-table/.

“Logarithm Rules – Chilimath”. 2021. Chilimath. https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/.

⭐ Lloyd, Philip. “What is the easiest way to understand logarithms?”. 2023. Quora. https://qr.ae/py66cZ.

Images

⭐ Budassi, Pablo Carlos. “A Logarithmic Map Of The Entire Observable Universe”. 2022. Visual Capitalist. https://www.visualcapitalist.com/cp/map-of-the-entire-known-universe/.