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Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:.

For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part.

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

A deeper study of logarithms requires the concept of a function. A function is a rule that, given one number, produces another number.

A proof of that fact requires the intermediate value theorem from elementary calculus. A function is continuous if it does not "jump", that is, if its graph can be drawn without lifting the pen.

The function that assigns to y its logarithm is called logarithm function or logarithmic function or just logarithm.

The formula for the logarithm of a power says in particular that for any number x ,. In prose, taking the x -th power of b and then the base- b logarithm gives back x.

Conversely, given a positive number y , the formula. Thus, the two possible ways of combining or composing logarithms and exponentiation give back the original number.

Inverse functions are closely related to the original functions. As a consequence, log b x diverges to infinity gets bigger than any given number if x grows to infinity, provided that b is greater than one.

In that case, log b x is an increasing function. Analytic properties of functions pass to their inverses. Roughly, a continuous function is differentiable if its graph has no sharp "corners".

It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the constant e.

The derivative with a generalised functional argument f x is. The quotient at the right hand side is called the logarithmic derivative of f.

Computing f' x by means of the derivative of ln f x is known as logarithmic differentiation. Related formulas , such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.

The right hand side of this equation can serve as a definition of the natural logarithm. Product and power logarithm formulas can be derived from this definition.

In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size.

Therefore, the left hand blue area, which is the integral of f x from t to tu is the same as the integral from 1 to u. This justifies the equality 2 with a more geometric proof.

It is closely tied to the natural logarithm : as n tends to infinity , the difference,. This relation aids in analyzing the performance of algorithms such as quicksort.

There are also some other integral representations of the logarithm that are useful in some situations:. The second identity can be proven by writing.

Almost all real numbers are transcendental. The logarithm is an example of a transcendental function. The Gelfond—Schneider theorem asserts that logarithms usually take transcendental, i.

In general, logarithms can be calculated using power series or the arithmetic—geometric mean , or be retrieved from a precalculated logarithm table that provides a fixed precision.

This is a shorthand for saying that ln z can be approximated to a more and more accurate value by the following expressions:. This series approximates ln z with arbitrary precision, provided the number of summands is large enough.

In elementary calculus, ln z is therefore the limit of this series. Another series is based on the area hyperbolic tangent function:.

This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if z is close to 1.

The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. A can be calculated using the exponential series , which converges quickly provided y is not too large.

A closely related method can be used to compute the logarithm of integers. The arithmetic—geometric mean yields high precision approximations of the natural logarithm.

Sasaki and Kanada showed in that it was particularly fast for precisions between and decimal places, while Taylor series methods were typically faster when less precision was needed.

Here M x , y denotes the arithmetic—geometric mean of x and y. The two numbers quickly converge to a common limit which is the value of M x , y.

A larger m makes the M x , y calculation take more steps the initial x and y are farther apart so it takes more steps to converge but gives more precision.

The constants pi and ln 2 can be calculated with quickly converging series. While at Los Alamos National Laboratory working on the Manhattan Project , Richard Feynman developed a bit-processing algorithm that is similar to long division and was later used in the Connection Machine.

Any base may be used for the logarithm table. Logarithms have many applications inside and outside mathematics.

Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor.

This gives rise to a logarithmic spiral. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.

Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log x grows very slowly for large x , logarithmic scales are used to compress large-scale scientific data.

Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation , the Fenske equation , or the Nernst equation.

Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a unit of measurement associated with logarithmic-scale quantities.

It is based on the common logarithm of ratios —10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio.

It is used to quantify the loss of voltage levels in transmitting electrical signals, [61] to describe power levels of sounds in acoustics , [62] 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.

The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale or the Richter magnitude scale.

For example, a 5. It measures the brightness of stars logarithmically. Vinegar typically has a pH of about 3.

Semilog log—linear graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically.

For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space on the vertical axis as the increase from 1 to 1 million.

This is applied in visualizing and analyzing power laws. Logarithms occur in several laws describing human perception : [69] [70] Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.

Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to as is to Increasing education shifts this to a linear estimate positioning 10 times as far away in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.

Logarithms arise in probability theory : the law of large numbers dictates that, for a fair coin , as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half.

The fluctuations of this proportion about one-half are described by the law of the iterated logarithm. Logarithms also occur in log-normal distributions.

When the logarithm of a random variable has a normal distribution , the variable is said to have a log-normal distribution.

Logarithms are used for maximum-likelihood estimation of parametric statistical models. For such a model, the likelihood function depends on at least one parameter that must be estimated.

The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables. Benford's law describes the occurrence of digits in many data sets , such as heights of buildings.

Auditors examine deviations from Benford's law to detect fraudulent accounting. Analysis of algorithms is a branch of computer science that studies the performance of algorithms computer programs solving a certain problem.

For example, to find a number in a sorted list, the binary search algorithm checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found.

This algorithm requires, on average, log 2 N comparisons, where N is the list's length. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.

A function f x is said to grow logarithmically if f x is exactly or approximately proportional to the logarithm of x. Biological descriptions of organism growth, however, use this term for an exponential function.

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.

In statistical thermodynamics , the entropy S of some physical system is defined as. The sum is over all possible states i of the system in question, such as the positions of gas particles in a container.

Moreover, p i is the probability that the state i is attained and k is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information.

If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log 2 N bits.

Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle.

Such systems are chaotic in a deterministic way, because small measurement errors of the initial state predictably lead to largely different final states.

Logarithms occur in definitions of the dimension of fractals. The Sierpinski triangle pictured can be covered by three copies of itself, each having sides half the original length.

Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question. Logarithms are related to musical tones and intervals.

In equal temperament , the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch , of the individual tones.

Accordingly, the frequency ratios agree:. The latter is used for finer encoding, as it is needed for non-equal temperaments.

Natural logarithms are closely linked to counting prime numbers 2, 3, 5, 7, 11, The logarithm of n factorial , n! This can be used to obtain Stirling's formula , an approximation of n!

All the complex numbers a that solve the equation. Such a number can be visualized by a point in the complex plane , as shown at the right. The polar form encodes a non-zero complex number z by its absolute value , that is, the positive, real distance r to the origin , and an angle between the real x axis Re and the line passing through both the origin and z.

This angle is called the argument of z. The absolute value r of z is given by. Euler's formula connects the trigonometric functions sine and cosine to the complex exponential :.

Using this formula, and again the periodicity, the following identities hold: [98]. Therefore, the complex logarithms of z , which are all those complex values a k for which the a k -th power of e equals z , are the infinitely many values.

The principal argument of any positive real number x is 0; hence Log x is a real number and equals the real natural logarithm. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.

This way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there.

This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i. Such a locus is called a branch cut.

Dropping the range restrictions on the argument makes the relations "argument of z ", and consequently the "logarithm of z ", multi-valued functions.

Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the multi-valued inverse function of the matrix exponential.

Both are defined via Taylor series analogous to the real case. Its inverse is also called the logarithmic or log map.

In the context of finite groups exponentiation is given by repeatedly multiplying one group element b with itself. The discrete logarithm is the integer n solving the equation.

Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography , such as for example in the Diffie—Hellman key exchange , a routine that allows secure exchanges of cryptographic keys over unsecured information channels.

Further logarithm-like inverse functions include the double logarithm ln ln x , the super- or hyperlogarithm a slight variation of which is called iterated logarithm in computer science , the Lambert W function , and the logit.

The "Appears As" column contains the text that appears in the log file. The data in the table is in the order of occurrence in the log file record.

The field prefixes in the file are defined as follows:. The application can select one or more of the W3C Extended log file fields, however, not all fields will contain information.

For fields that are selected but for which there is no information, a hyphen - appears as a placeholder. Fields are separated by spaces.

If a field is enabled by the URL group or server session, but not selected for the request, it appears in the log file with a hyphen - as a placeholder.

Log files are created when the first request arrives on the URL Group or server session, they are not created when logging is configured.

The time-taken timestamp is stopped when the last send completion occurs. Time-taken does not reflect time across the network.

The first request to the site shows a slightly longer time taken than other similar requests because the HTTP Server API opens the log file with the first request.

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