Logarithm

In mathematics, the logarithm of a number tells us the power to which we must raise a fixed number, called the base, to get that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 equals 10 raised to the 3rd power: 1000 = 10³ = 10 × 10 × 10.

The logarithm with base 10, called the decimal or common logarithm, is often used in science and engineering. Another important type is the natural logarithm, which uses the number e ≈ 2.718 as its base. The binary logarithm uses base 2 and is important in computer science, information theory, music theory, and photography.

Plots of logarithm functions, with three commonly used bases. The special points logb b = 1 are indicated by dotted lines, and all curves intersect in logb 1 = 0.

Logarithms were created by John Napier to make complex calculations easier. They were quickly used by navigators, scientists, engineers, surveyors, and others. With logarithm tables, difficult multiplications could be turned into simpler additions. Later, Leonhard Euler connected logarithms to the exponential function.

Logarithms are also used in special scales that make very large or small numbers easier to work with. For example, the decibel (dB) is used to measure sound levels, and pH in chemistry measures how acidic a solution is. They appear in many scientific formulae, and in measuring the complexity of algorithms.

Motivation

The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) = 3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but does not meet it.

Addition, multiplication, and exponentiation are three important math actions. The opposite of addition is subtraction, and the opposite of multiplication is division. In the same way, a logarithm is the opposite action of exponentiation.

Exponentiation is when a number, called the base, is raised to a power, called the exponent, to get another value. For example, raising 2 to the power of 3 gives 8, because 2 × 2 × 2 = 8. The logarithm tells us what exponent we need to use to get from the base to the number.

Definition

The logarithm of a number tells us how many times we need to multiply another number (called the base) to get that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 equals 10 multiplied by itself 3 times: 10 × 10 × 10 = 1000.

We write this as log₁₀ 1000 = 3. In general, if we have a number x and a base b, the logarithm of x to base b is the number y such that when we raise b to the power of y, we get x. So, if x = bʸ, then y is the logarithm of x to base b.

Logarithmic identities

Main article: List of logarithmic identities

Logarithms have special rules that make them easier to use. One rule says that when you multiply two numbers, the logarithm of the product is the sum of their logarithms. Another rule says that when you divide two numbers, the logarithm of the quotient is the difference of their logarithms.

There is also a rule for powers: the logarithm of a number raised to a power is that power times the logarithm of the number. For roots, the logarithm of a root is the logarithm of the number divided by the root’s index. These rules help make working with logarithms easier.

Product, quotient, power, and root identities of logarithms
Identity	Formula	Example
Product	log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y {\textstyle \log _{b}(xy)=\log _{b}x+\log _{b}y}	log 3 ⁡ 243 = log 3 ⁡ ( 9 ⋅ 27 ) = log 3 ⁡ 9 + log 3 ⁡ 27 = 2 + 3 = 5 {\textstyle \log _{3}243=\log _{3}(9\cdot 27)=\log _{3}9+\log _{3}27=2+3=5}
Quotient	log b x y = log b ⁡ x − log b ⁡ y {\textstyle \log _{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y}	log 2 ⁡ 16 = log 2 64 4 = log 2 ⁡ 64 − log 2 ⁡ 4 = 6 − 2 = 4 {\textstyle \log _{2}16=\log _{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}
Power	log b ⁡ ( x p ) = p log b ⁡ x {\textstyle \log _{b}\left(x^{p}\right)=p\log _{b}x}	log 2 ⁡ 64 = log 2 ⁡ ( 2 6 ) = 6 log 2 ⁡ 2 = 6 {\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log _{2}2=6}
Root	log b ⁡ x p = log b ⁡ x p {\textstyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}}	log 10 ⁡ 1000 = 1 2 log 10 ⁡ 1000 = 3 2 = 1.5 {\textstyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}

Particular bases

Three bases are very common in math: 10, e (a special number about 2.718), and 2. The base e is often used in advanced math because it has useful properties. Base 10 is easy to work with when doing calculations by hand, especially with decimal numbers.

For example, the base 10 logarithm of a number tells us how many digits are in that number. If we know the logarithm, we can find the number of digits by rounding up to the next whole number. Both the base e logarithm and the base 2 logarithm are important in information theory. They help us measure information in different units. Base 2 is also used in computer science because computers use binary numbers, and in music to measure pitch differences.

Sometimes, when we just write "log" without saying the base, the base depends on the subject we are working with. In science and engineering, it often means base 10. In pure math, it usually means base e. In computer science and information theory, it often means base 2.

Base b	Name for log_b x	ISO notation	Other notations
2	binary logarithm	lb x	ld x, log x, lg x, log₂ x
e	natural logarithm	ln x	log x, log_e x
10	common logarithm	lg x	log x, log₁₀ x
b	logarithm to base b	log_b x

History

Main article: History of logarithms

In the 1600s, mathematicians in Europe found a new way to work with numbers. John Napier discovered this in 1614. Before this, people used other methods to make math easier. Napier used the word "logarithmus," meaning 'ratio-number'.

The common logarithm helps us understand how many times we need to multiply ten to reach a certain number. This idea was also discussed long ago by Archimedes. These new methods turned multiplication into addition, which made calculations faster. Some methods used tables based on angles and shapes.

Later, a mathematician named Grégoire de Saint-Vincent worked on a problem about a special curve called a hyperbola. His work led to what we now call the natural logarithm. This idea was liked by famous mathematicians like Christiaan Huygens and James Gregory. Another mathematician, Gottfried Wilhelm Leibniz, created a special symbol for it in 1675.

Before modern ideas developed, Roger Cotes found an important connection in 1714.

Logarithm tables, slide rules, and historical applications

Logarithms made hard calculations easier before calculators and computers existed. They helped scientists, especially in astronomy, surveying, and celestial navigation. One famous scientist, Pierre-Simon Laplace, said logarithms saved time and reduced mistakes in long calculations.

The 1797 Encyclopædia Britannica explanation of logarithms

A key tool was the table of logarithms. The first table was made by Henry Briggs in 1617. It listed logarithms for numbers from 1 to 1000 very accurately. These tables helped people do multiplication and division faster by adding and subtracting logarithms instead.

Log tables

Tables listed the values of logarithms for different numbers. This made calculations much quicker than older methods. For example, to multiply two numbers, people would add their logarithms and then find the result using the table again.

Computations

With logarithms, multiplying or dividing numbers became as easy as adding or subtracting their logarithms. This saved time and reduced errors in calculations.

Slide rules

Main article: Slide rule

Another useful tool was the slide rule. It used moving scales to add logarithms mechanically. For example, aligning the numbers 2 and 3 would show the product 6. Slide rules were important tools for engineers and scientists until calculators became common, because they allowed quick calculations even if they were not perfectly precise.

Analytic properties

Logarithms are connected to functions, which are rules that take one number and give back another. For example, a function can raise a number to the power of another number. This helps us understand how logarithms work.

A logarithm tells us what power we need to raise a number, called the base, to get another number. For example, the logarithm of 1000 with base 10 is 3, because 10 raised to the 3rd power equals 1000. This idea helps us solve many problems in math and science.

Calculation

Logarithms are easy to figure out in some cases. For example, the logarithm of 1000 to base 10 is 3. In general, logarithms can be worked out using special math methods or by looking them up in a book. There are also clever ways to calculate logarithms using just adding and shifting numbers.

Power series

The logarithm keys (LOG for base 10 and LN for base e) on a TI-83 Plus graphing calculator

Taylor series

Main article: Mercator series

For certain numbers, we can use a special math pattern to find the logarithm. This pattern works well when the number is close to 1. For example, we can use this pattern to find the logarithm of 1.5. This method gets better as we use more parts of the pattern.

The Taylor series of ln(z) centered at z = 1. The animation shows the first 10 approximations along with the 99th and 100th. The approximations do not converge beyond a distance of 1 from the center.

Arithmetic–geometric mean approximation

The arithmetic–geometric mean can give very exact answers for the natural logarithm. A method from 1982 shows how to use this to get precise results. This method uses a special formula to get closer to the true value of the logarithm.

Feynman's algorithm

While working on important science projects, a scientist named Richard Feynman created a way to compute logarithms. This method builds up the answer by adding small pieces together.

Applications

A nautilus shell displaying a logarithmic spiral

Logarithms have many uses in math and science. They help us understand patterns that repeat at different sizes, like the chambers of a nautilus shell. They also help us measure very large or very small changes more easily.

In science, logarithms help us describe things like how loud a sound is or how strong an earthquake is. They are used to measure the brightness of stars and the acidity of liquids in chemistry. These special scales make it easier to work with numbers that vary a lot in size.

Interval (the two tones are played at the same time)	1/12 tone play^ⓘ	Semitone play^ⓘ	Just major third play^ⓘ	Major third play^ⓘ	Tritone play^ⓘ	Octave play^ⓘ
Frequency ratio r {\displaystyle r}	2 1 72 ≈ 1.0097 {\displaystyle 2^{\frac {1}{72}}\approx 1.0097}	2 1 12 ≈ 1.0595 {\displaystyle 2^{\frac {1}{12}}\approx 1.0595}	5 4 = 1.25 {\displaystyle {\tfrac {5}{4}}=1.25}	2 4 12 = 2 3 ≈ 1.2599 {\displaystyle {\begin{aligned}2^{\frac {4}{12}}&={\sqrt[{3}]{2}}\\&\approx 1.2599\end{aligned}}}	2 6 12 = 2 ≈ 1.4142 {\displaystyle {\begin{aligned}2^{\frac {6}{12}}&={\sqrt {2}}\\&\approx 1.4142\end{aligned}}}	2 12 12 = 2 {\displaystyle 2^{\frac {12}{12}}=2}
Number of semitones 12 log 2 ⁡ r {\displaystyle 12\log _{2}r}	1 6 {\displaystyle {\tfrac {1}{6}}}	1 {\displaystyle 1}	≈ 3.8631 {\displaystyle \approx 3.8631}	4 {\displaystyle 4}	6 {\displaystyle 6}	12 {\displaystyle 12}
Number of cents 1200 log 2 ⁡ r {\displaystyle 1200\log _{2}r}	16 2 3 {\displaystyle 16{\tfrac {2}{3}}}	100 {\displaystyle 100}	≈ 386.31 {\displaystyle \approx 386.31}	400 {\displaystyle 400}	600 {\displaystyle 600}	1200 {\displaystyle 1200}

Generalizations

Complex logarithm

Main article: Complex logarithm

Complex numbers are special numbers written as z = x + iy, where x and y are real numbers and i is the imaginary unit. This means i squared equals -1. These numbers can be shown as points on a graph called the complex plane.

The complex logarithm finds all possible answers to the equation e^a = z, where z is a complex number. This gives many values, based on how many times you "wrap around" the origin when finding the angle of the complex number.

One special value, called the principal value, is chosen by limiting the angle to a specific range. This creates a jump along the negative real axis, called a branch cut. Without this limit, the logarithm would have many values.

Inverses of other exponential functions

Exponential functions are used in many areas of math, and their inverses are often called logarithms. For example, the logarithm of a matrix is the inverse of the matrix exponential. In group theory, the discrete logarithm solves the equation bⁿ = x, where b and x are elements of a finite group. Finding this n can be very hard in some groups.

Other logarithm-like functions include the double logarithm, the iterated logarithm, the Lambert W function, and the logit function. These are inverses of other special functions.

Related concepts

In group theory, the logarithm changes multiplication into addition. This is a special mapping called an isomorphism between positive real numbers under multiplication and real numbers under addition.

Logarithmic forms also appear in complex analysis and algebraic geometry. The polylogarithm function is defined as a sum, and for a special case, it relates to the natural logarithm and the Riemann zeta function.