Logarithm

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.

Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography.

(If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)

by which tables of logarithms allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers.

[8] Binary logarithms are also used in computer science, where the binary system is ubiquitous; in music theory, where a pitch ratio of two (the octave) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional equal temperament), or equivalently the log base 21/1200  ; and in photography rescaled base 2 logarithms are used to measure exposure values, light levels, exposure times, lens apertures, and film speeds in "stops".

[13] The history of logarithms in seventeenth-century Europe saw the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods.

[19][20] Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600.

[21][22] Napier coined the term for logarithm in Middle Latin, logarithmus, literally meaning 'ratio-number', derived from the Greek logos 'proportion, ratio, word' + arithmos 'number'.

[24] The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation.

Invention of the function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague.

Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that[27]

Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.

William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other.

The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.

It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range.

That is, the slope of the tangent touching the graph of the base-b logarithm at the point (x, logb (x)) equals 1/(x ln(b)).

[48][49] Moreover, the binary logarithm algorithm calculates lb(x) recursively, based on repeated squarings of x, taking advantage of the relation

The nth partial sum can approximate ln(z) with arbitrary precision, provided the number of summands n is large enough.

While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman developed a bit-processing algorithm to compute the logarithm that is similar to long division and was later used in the Connection Machine.

It is used to quantify the attenuation or amplification of electrical signals,[57] to describe power levels of sounds in acoustics,[58] and the absorbance of light in the fields of spectrometry and optics.

The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels.

[59] In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression methods using the logarithm.

In such graphs, exponential functions of the form f(x) = a · bx appear as straight lines with slope equal to the logarithm of b. Log-log graphs scale both axes logarithmically, which causes functions of the form f(x) = a · xk to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing power laws.

[69] In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation such as the actual vs. the perceived weight of an item a person is carrying.

According to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals log10 (d + 1) − log10 (d), regardless of the unit of measurement.

In other words, the amount of memory needed to store N grows logarithmically with N. Entropy is broadly a measure of the disorder of some system.

[88] Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure.

Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.

The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing π(x) and Li(x).

Plots of logarithm functions, with three commonly used bases. The special points log b b = 1 are indicated by dotted lines, and all curves intersect in log b 1 = 0 .
Graph showing a logarithmic curve, crossing the x-axis at x= 1 and approaching minus infinity along the y-axis.
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., log 2 (8) = 3 and 2 3 = 8 . The graph gets arbitrarily close to the y -axis, but does not meet it .
Overlaid graphs of the logarithm for bases ⁠ 1 / 2 ⁠ , 2, and e
The 1797 Encyclopædia Britannica explanation of logarithms
A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6.
Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x .
The graphs of two functions.
The graph of the logarithm function log b ( x ) (blue) is obtained by reflecting the graph of the function b x (red) at the diagonal line ( x = y ).
A graph of the logarithm function and a line touching it in one point.
The graph of the natural logarithm (green) and its tangent at x = 1.5 (black)
A hyperbola with part of the area underneath shaded in grey.
The natural logarithm of t is the shaded area underneath the graph of the function f ( x ) = 1/ x .
The hyperbola depicted twice. The area underneath is split into different parts.
A visual proof of the product formula of the natural logarithm
The logarithm keys (LOG for base 10 and LN for base e ) on a TI-83 Plus graphing calculator
An animation showing increasingly good approximations of the logarithm graph.
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.
A photograph of a nautilus' shell.
A nautilus shell displaying a logarithmic spiral
A graph of the value of one mark over time. The line showing its value is increasing very quickly, even with logarithmic scale.
A logarithmic chart depicting the value of one Goldmark in Papiermarks during the German hyperinflation in the 1920s
Three asymmetric PDF curves
Three probability density functions (PDF) of random variables with log-normal distributions. The location parameter μ , which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.
A bar chart and a superimposed second chart. The two differ slightly, but both decrease in a similar fashion.
Distribution of first digits (in %, red bars) in the population of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law.
An oval shape with the trajectories of two particles.
Billiards on an oval billiard table . Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of reflections at the boundary.
Parts of a triangle are removed in an iterated way.
The Sierpinski triangle (at the right) is constructed by repeatedly replacing equilateral triangles by three smaller ones.
An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis.
Polar form of z = x + iy . Both φ and φ' are arguments of z .
A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.
The principal branch (- π , π ) of the complex logarithm, Log( z ) . The black point at z = 1 corresponds to absolute value zero and brighter colors refer to bigger absolute values. The hue of the color encodes the argument of Log( z ) .