The inverse of the exponential: given a number, it tells you the exponent that produces it from the base.

Who invented logarithms, and when?

John Napier, who published them in 1614 in Mirifici Logarithmorum Canonis Descriptio.

Why did Napier invent logarithms?

To save the labor of hand calculation with large numbers: they turn multiplication and division into addition and subtraction.

The time it takes for half the radioactive nuclei in a sample to decay; for carbon-14 it is 5,730 years.

How much does seismic wave amplitude change between two consecutive magnitudes on the Richter scale?

Roughly 10 times, according to the USGS: each extra unit on the scale corresponds to a factor of 10 in amplitude.

What does pH measure, mathematically?

The negative base-10 logarithm of the hydrogen ion concentration in a solution: one extra pH unit means the concentration changed tenfold.

Why is the decibel a logarithmic unit?

Because OSHA figures show the scale barely moves when the physical sound power doubles — only 3 dB — so small changes in dB hide large changes in sound pressure.

Logarithms Explained: What Exponentials and Logarithms Are Really For

In 30 seconds quick read

An exponential raises a fixed base to a power that changes; a logarithm runs the same path backward, asking which power the base must reach to produce a given number. John Napier published logarithms in 1614 to replace tedious multiplication and division with simple addition. They describe phenomena that grow or shrink by a constant factor, such as radioactive decay, and they read scales that squeeze huge numbers into a handful of steps, such as earthquake magnitude, pH and decibels. The link between the two operations is why they keep showing up together, in physics, chemistry and acoustics.

Key Points

A logarithm answers the question "to what power must I raise the base to get this number": it is the inverse of the exponential operation.
John Napier published logarithms in 1614 to turn tedious multiplication and division into simple addition and subtraction.
Radioactive decay follows an exponential law, N = N0 · e^(-λt): the number of remaining nuclei falls over time from the starting number.
The Richter scale is logarithmic: according to the USGS, each extra unit of magnitude corresponds to seismic waves with 10 times the amplitude.
pH measures acidity with a negative base-10 logarithm: each unit of pH corresponds to a factor of 10 in hydrogen ion concentration.
The decibel is a logarithmic unit: doubling the physical sound power, according to OSHA, adds only 3 dB to the scale.

Key figures

1614 The year John Napier published Mirifici Logarithmorum Canonis Descriptio, the work that introduced logarithms. Source: MacTutor History of Mathematics
1935 The year Richter and Gutenberg defined the original earthquake magnitude relation, for local earthquakes. Source: USGS, Magnitude Types
4.5 billion years The half-life of uranium-238, used to date the oldest materials on the planet. Source: Lumen Learning / OpenStax, College Physics

Deep Dive

Two sides of the same function

An exponential starts from a fixed base and an exponent that changes: 2 raised to x doubles the result with every extra step on the exponent. A logarithm runs the opposite path: it asks which power the base must reach to arrive at a given number. If 2 cubed is 8, the logarithm of 8 in base 2 is 3. Applied one after the other, the two operations cancel each other out.

That link is not an exercise detail. It is the reason a logarithm shows up whenever something grows or shrinks by multiplying by the same factor at each step, rather than by adding the same amount every time.

Napier and the burden of calculation

John Napier published logarithms in 1614, in the Mirifici Logarithmorum Canonis Descriptio, as the MacTutor History of Mathematics biography of Napier records. Napier explained that he had devised them to remove the burden of calculating with large numbers: multiplication, division and root extraction.

The idea was to replace numbers that require multiplication and division with other numbers — logarithms — that reach the same result through plain addition and subtraction. The symbolic manipulation that makes those numbers workable is the same groundwork covered in Algebra: What It Is and How Symbolic Notation Works.

When growth is a factor, not a sum

According to the physics notes of College Physics (Lumen Learning/OpenStax), radioactive decay follows an exponential law: N = N0 · e^(-λt), where the number of remaining nuclei N falls over time from the starting number N0. The half-life is the time it takes for half the nuclei in a sample to decay.

Practical example: carbon-14 has a half-life of 5,730 years. An organic sample that retains half the amount it held while alive corresponds, on average, to 5,730 years of decay. Uranium-238 has a half-life of roughly 4.5 billion years, and it is one of the tools used to date the oldest materials on the planet.

The logarithm enters here as a way to invert the exponential: knowing how much radioactive material is left, it lets you work out the elapsed time directly, instead of testing one value after another. Isolating an unknown sitting in the exponent, step by step, is the technique covered in Exponential Equations: How to Solve Them, With Worked Examples.

When the scale itself is logarithmic

Some measurement scales are built as logarithms from the start, to compress numbers that would otherwise range from tiny fractions to billions.

For earthquakes, according to the USGS Yellowstone Volcano Observatory, each extra unit on the magnitude scale corresponds to seismic waves with 10 times the amplitude: a magnitude-4 earthquake produces waves 10 times the amplitude of a magnitude-3 one. The original magnitude relation was defined by Richter and Gutenberg in 1935, for local earthquakes. In terms of energy released, rather than amplitude alone, USGS publications put a magnitude-7 earthquake at roughly 30 times the energy of a magnitude-6 one, and a magnitude-8 earthquake at roughly 900 times as much: the effect multiplies by about 30 for every unit of magnitude. The mechanism behind those waves is covered in Volcanoes and Earthquakes: Why the Earth Shakes.

pH works the same way. According to the EPA’s CADDIS, pH is the negative base-10 logarithm of the hydrogen ion concentration in an aqueous solution: each one-unit change corresponds to a factor of 10 in concentration, so a two-unit difference means a factor of 100.

The decibel is also a logarithmic unit. OSHA notes that a small change in the decibel count corresponds to a large change in sound pressure: doubling the physical sound power adds only 3 dB to the scale, and sound pressure drops by 6 dB every time the distance from a point source doubles.

Scale	What it measures	Factor per unit
Richter, amplitude	seismic waves	×10
Richter, energy	seismic waves	×30 (≈×900 over two units)
pH	hydrogen ion concentration	×10
Decibel	sound pressure	+3 dB = double the power

Where they meet again

Exponentials and logarithms belong to the toolkit covered in Calculus: What It Is and How It Works, alongside limits and derivatives. The Richter scale, pH and the decibel share the same structure: a small number on the scale corresponds to a large factor in the underlying quantity it describes.

Common myths

✗ Myth Many people treat logarithms and exponentials as two separate topics, each to be studied on its own.

✓ Reality A logarithm is the inverse of the exponential: an exponential turns a power into a number, and a logarithm works backward from that number to the power that produced it.
✗ Myth A common assumption is that a logarithmic scale, like the one used for earthquakes or decibels, grows in a straight line, so that double the number means double the effect.

✓ Reality Each added unit multiplies the effect by a fixed factor instead. The USGS puts the energy released by a magnitude-8 earthquake at roughly 900 times that of a magnitude-6 one, not twice as much.
✗ Myth Logarithms are sometimes assumed to be a product of calculators or computers.

✓ Reality They date back to 1614, when John Napier published them to remove the burden of hand calculation with large numbers, centuries before any electronics existed.

Mind map

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Quiz: test yourself

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Grade 0/10 0/5

1 What is a logarithm, in simple terms?

2 Why did John Napier publish logarithms in 1614?

3 On the Richter scale, a magnitude-4 earthquake produces seismic waves roughly 10 times the amplitude of a magnitude-3 one. True or false?

4 What does the pH of a solution describe, mathematically?

5 Which element has a half-life of about 4.5 billion years, used to date the oldest rocks on Earth?

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Frequently asked questions

What is the difference between an exponential and a logarithm?

They are inverse operations. The exponential starts from a base and an exponent and returns a number; the logarithm starts from that number and the base, and returns the exponent that connects them.

What are logarithms really used for in real life?

They read quantities that vary over enormous ranges with a compressible scale. Earthquake magnitude, the pH of solutions and sound decibels are all logarithmic scales, built to make numbers manageable when the smallest and largest values would otherwise differ by dozens of digits.

What is an exponential function, in practice?

A function of the form a raised to the x, where the variable x sits in the exponent: it describes growth, or decay if the base is smaller than 1, where each step multiplies by the same fixed factor.

How do you solve an exponential equation using logarithms?

You apply a logarithm to both sides of the equation, moving the unknown from the exponent into a multiplying factor. The concrete steps, with worked examples, are in the Recap on Exponential Equations (/en/science/exponential-equations/).

Why are logarithms and exponentials called inverse functions?

Because applying them in sequence, in the right order, cancels the effect out: the logarithm in base b of b raised to x simply returns x, and the same works in reverse.