What is a logarithm? The opposite of an exponent, explained

A logarithm is the answer to the question "what power do I raise this base to, to get this number?" Since 10 × 10 × 10 = 1,000, the logarithm of 1,000 in base 10 is 3. A logarithm undoes an exponent, the way subtraction undoes addition. In practice it counts how many times a number has been multiplied by the base, which is why logarithms turn huge ranges into small, manageable ones.

In short

• log_b(x) = y means exactly the same as b^y = x: the logarithm is the missing exponent.
• Logarithms turn multiplication into addition, which is why they were invented, to make hard sums quick.
• On a log scale each step means "times ten" rather than "plus one", so a small difference in the number is a big difference in the thing measured.

What does a logarithm actually tell you?

It tells you an exponent. The notation is log_b(x), read "log base b of x", where b is the base and x is the number you start with. The statement log_b(x) = y says the same thing as b^y = x, just turned around. So log₁₀(1,000) = 3 because 10³ = 1,000, and log₂(64) = 6 because 2⁶ = 64.

A useful way to see the second one is to count halvings. Start at 64 and halve until you reach 1: 32, 16, 8, 4, 2, 1. That took six steps, so log₂(64) = 6. Logarithms count repeated multiplication the way division counts repeated subtraction.

A few values follow straight from the definition. The log of 1 is 0 in every base, because any base to the power 0 is 1. The log of the base itself is 1. Numbers between 0 and 1 have negative logs: log₁₀(0.01) = −2, because 10⁻² is one hundredth. And the log of 0 or of a negative number isn't defined, since no power of a positive base ever reaches them.

In base 10, the whole-number part of a log is the count of digits minus one. log₁₀(5,000) is about 3.7, and 5,000 has four digits. That is the everyday intuition: a base-10 log measures how big a number is in powers of ten, its order of magnitude.

Which bases do people use?

Three bases cover nearly everything. The common logarithm, base 10, is written log or log₁₀ and goes with the decimal system and with scientific scales. The binary logarithm, base 2, goes with doubling and with computers: picking one item out of a million takes 20 yes-or-no questions, because log₂(1,000,000) is just under 20. The natural logarithm, written ln, uses the base e, about 2.718, and is the one that appears in calculus and in growth and decay. Any of them can be turned into another by dividing: log₂(x) = log₁₀(x) ÷ log₁₀(2).

Why do the rules of logarithms work?

The rules aren't separate facts to memorize; each one is an exponent rule read backward. Multiplying powers of the same base adds their exponents: 10² × 10³ = 10⁵, because 100 × 1,000 = 100,000. Since logs are exponents, that becomes the product rule:

• Product: log(a × b) = log(a) + log(b).
• Quotient: log(a ÷ b) = log(a) − log(b), because dividing powers subtracts exponents.
• Power: log(a^n) = n × log(a), because raising a power to a power multiplies exponents.

The product rule is the reason logarithms were invented. Here it is at work, multiplying 2,000 by 3,000 the way a 17th-century astronomer would have, with a table of logs:

1. Look up log₁₀(2,000) = 3.3010 and log₁₀(3,000) = 3.4771.
2. Add them: 3.3010 + 3.4771 = 6.7781. A hard multiplication has become an easy addition.
3. Look up which number has a log of 6.7781. The table says 6,000,000.

With seven-digit numbers, that saves a great deal of work and many errors. A logarithm turns multiplying into adding, and powers into multiplying. A common slip runs the other way: log(a + b) is not log(a) + log(b), because no exponent rule says anything about adding.

Logarithms also solve the problem algebra gets stuck on, an unknown in the exponent. How long does something growing 7% a year take to double? That means solving 1.07^t = 2. Take logs of both sides and the power rule brings t down: t × log(1.07) = log(2), so t = 0.3010 ÷ 0.0294, about 10.2 years. That is where the shortcut for doubling time in exponential growth comes from.

Who invented logarithms, and what was the slide rule?

The Scottish landowner John Napier published logarithms in 1614, in a book called Mirifici logarithmorum canonis descriptio, the description of the wonderful canon of logarithms. Mathematics was his hobby. The English mathematician Henry Briggs visited him in 1615, and the two agreed on tables built on base 10 with the log of 1 set to 0, the common logarithms that Briggs then computed, according to the MacTutor History of Mathematics archive. Astronomers adopted them at once; a remark often attributed to Pierre-Simon Laplace says logarithms, by shortening the labor, doubled the life of the astronomer.

The tables soon became a tool you could hold. In 1620 Edmund Gunter marked a logarithmic scale along a 2-foot ruler, so that multiplying meant adding two lengths with a pair of dividers. William Oughtred then slid two of Gunter's rulers against each other, doing away with the dividers, and also designed a circular slide rule; his descriptions were published in 1632, MacTutor records. Slide rules, which add lengths proportional to logarithms, did the arithmetic of engineering for more than 300 years, until electronic pocket calculators replaced them in the 1970s.

Why do decibels, pH and earthquakes use logarithmic scales?

Because the things they measure range over many powers of ten, and a log scale squeezes that range into numbers people can say. Each step up a log scale multiplies the quantity rather than adding to it.

Scale	What it measures	One step up means	Example
Decibels (dB)	Sound level, 10 × log₁₀ of a power ratio	10 dB is 10 times the power; 3 dB is about double	Conversation is 60–70 dBA; long exposure at 85 dBA or more can damage hearing
pH	Acidity, minus log₁₀ of hydrogen-ion concentration	1 unit lower is 10 times more acidic	Lemon juice near pH 2 is about 100,000 times more acidic than pure water at pH 7
Earthquake magnitude	Size of the quake: two-thirds of log₁₀ of its seismic moment, plus a constant	1 unit is about 32 times the energy; 2 units, 1,000 times	A magnitude 8 releases about 1,000 times the energy of a magnitude 6

The hearing figures are from the US National Institute on Deafness and Other Communication Disorders, and the pH scale has its own note. The trap with log scales is reading them like ordinary ones. A jump from 60 to 90 decibels looks like half as much again, but it is 1,000 times the sound power.

Human senses work roughly this way too: equal steps in perceived loudness or brightness correspond to multiplying the physical signal, which is one reason decibels feel natural. And in physics, Ludwig Boltzmann's formula for entropy takes the logarithm of the number of ways a system can be arranged, so that counts with trillions of digits become numbers that can be added.

Check yourself

• Without looking back, explain why log₂(64) is 6, by counting halvings.
• Which exponent rule gives the product rule for logarithms, and how did that make multiplication easier before calculators?
• A sound rises from 50 to 70 decibels. How many times greater is its power, and why?

Sources

• John Napier, biography — MacTutor History of Mathematics, University of St Andrews
• William Oughtred, biography — MacTutor History of Mathematics, University of St Andrews
• Logarithmic functions — OpenStax, College Algebra 2e, Rice University, 2021
• Noise-induced hearing loss — National Institute on Deafness and Other Communication Disorders, NIH
• Moment magnitude scale — Wikipedia