Logarithm

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$$\backslash frac\{d\}\{dx\}\; \backslash log\_b\; x\; =\; \backslash frac\{1\}\{x\backslash ln\; b\}.$$

That is, the slope of the tangent touching the graph of the logarithm at the point equals .

The derivative of is ; this implies that is the unique antiderivative of that has the value 0 for . 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 .

The derivative with a generalized functional argument is

$$\backslash frac\{d\}\{dx\}\; \backslash ln\; f(x)\; =\; \backslash frac\{f\text{'}(x)\}\{f(x)\}.$$

The quotient at the right hand side is called the logarithmic derivative of . Computing by means of the derivative of is known as logarithmic differentiation., p. 386 The antiderivative of the natural logarithm is:

$$\backslash int\; \backslash ln(x)\; \backslash ,dx\; =\; x\; \backslash ln(x)\; -\; x\; +\; C.$$

Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.

$$\backslash ln\; t\; =\; \backslash int\_1^t\; \backslash frac\{1\}\{x\}\; \backslash ,\; dx.$$

This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, equals the area between the -axis and the graph of the function , ranging from to . This is a consequence of the fundamental theorem of calculus and the fact that the derivative of is . Product and power logarithm formulas can be derived from this definition., section III.6 For example, the product formula is deduced as:

The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (). 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  and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function again. Therefore, the left hand blue area, which is the integral of from to is the same as the integral from 1 to . This justifies the equality (2) with a more geometric proof.

The power formula may be derived in a similar way:

The second equality uses a change of variables (integration by substitution), .

The sum over the reciprocals of natural numbers,

$$1\; +\; \backslash frac\; 1\; 2\; +\; \backslash frac\; 1\; 3\; +\; \backslash cdots\; +\; \backslash frac\; 1\; n\; =\; \backslash sum\_\{k=1\}^n\; \backslash frac\{1\}\{k\},$$

is called the harmonic series. It is closely tied to the natural logarithm: as tends to infinity, the difference,

$$\backslash sum\_\{k=1\}^n\; \backslash frac\{1\}\{k\}\; -\; \backslash ln(n),$$

converges (i.e. gets arbitrarily close) to a number known as the Euler–Mascheroni constant . This relation aids in analyzing the performance of algorithms such as quicksort., sections 11.5 and 13.8

$$\backslash log\_2\backslash left(x^2\backslash right)\; =\; 2\; \backslash log\_2\; |x|.$$

Equating the function to this infinite sum (series) is shorthand for saying that the function can be approximated to a more and more accurate value by the following expressions (known as ):

$$(z-1),\backslash \; \backslash \; (z-1)\; -\; \backslash frac\{(z-1)^2\}\{2\},\backslash \; \backslash \; (z-1)\; -\; \backslash frac\{(z-1)^2\}\{2\}\; +\; \backslash frac\{(z-1)^3\}\{3\},\backslash \; \backslash ldots$$

For example, with the third approximation yields , which is about greater than , and the ninth approximation yields , which is only about greater. The th partial sum can approximate with arbitrary precision, provided the number of summands is large enough.

In elementary calculus, the series is said to converge to the function , and the function is the limit of the series. It is the Taylor series of the natural logarithm at . The Taylor series of provides a particularly useful approximation to when is small, , since then $$\backslash ln\; (1+z)\; =\; z\; -\; \backslash frac\{z^2\}\{2\}\; +\backslash frac\{z^3\}\{3\}\; -\backslash cdots\; \backslash approx\; z.$$

For example, with the first-order approximation gives , which is less than off the correct value .

$$\backslash ln\; (z)\; =\; 2\backslash cdot\backslash operatorname\{artanh\}\backslash ,\backslash frac\{z-1\}\{z+1\}\; =\; 2\; \backslash left\; (\; \backslash frac\{z-1\}\{z+1\}\; +\; \backslash frac\{1\}\{3\}\{\backslash left(\backslash frac\{z-1\}\{z+1\}\backslash right)\}^3\; +\; \backslash frac\{1\}\{5\}\{\backslash left(\backslash frac\{z-1\}\{z+1\}\backslash right)\}^5\; +\; \backslash cdots\; \backslash right\; ),$$

for any real number . Using sigma notation, this is also written as

$$\backslash ln\; (z)\; =\; 2\backslash sum\_\{k=0\}^\backslash infty\backslash frac\{1\}\{2k+1\}\backslash left(\backslash frac\{z-1\}\{z+1\}\backslash right)^\{2k+1\}.$$

This series can be derived from the above Taylor series. It converges quicker than the Taylor series, especially if is close to 1. For example, for , the first three terms of the second series approximate with an error of about . The quick convergence for close to 1 can be taken advantage of in the following way: given a low-accuracy approximation and putting

$$A\; =\; \backslash frac\; z\{\backslash exp(y)\},$$

the logarithm of is:

$$\backslash ln\; (z)=y+\backslash ln\; (A).$$

The better the initial approximation is, the closer is to 1, so its logarithm can be calculated efficiently. can be calculated using the exponential series, which converges quickly provided is not too large. Calculating the logarithm of larger can be reduced to smaller values of by writing , so that .

A closely related method can be used to compute the logarithm of integers. Putting $\backslash textstyle\; z=\backslash frac\{n+1\}\{n\}$ in the above series, it follows that:

$$\backslash ln\; (n+1)\; =\; \backslash ln(n)\; +\; 2\backslash sum\_\{k=0\}^\backslash infty\backslash frac\{1\}\{2k+1\}\backslash left(\backslash frac\{1\}\{2\; n+1\}\backslash right)^\{2k+1\}.$$

If the logarithm of a large integer  is known, then this series yields a fast converging series for , with a rate of convergence of $\backslash left(\backslash frac\{1\}\{2\; n+1\}\backslash right)^\{2\}$.

$$\backslash ln\; (x)\; \backslash approx\; \backslash frac\{\backslash pi\}\{2\backslash ,\; \backslash mathrm\{M\}\backslash !\backslash left(1,\; 2^\{2\; -\; m\}/x\; \backslash right)\}\; -\; m\; \backslash ln(2).$$

Here denotes the arithmetic–geometric mean of and . It is obtained by repeatedly calculating the average (arithmetic mean) and $\backslash sqrt\{xy\}$ (geometric mean) of and then let those two numbers become the next and . The two numbers quickly converge to a common limit which is the value of . is chosen such that

$$x\; \backslash ,2^m\; >\; 2^\{p/2\}.\backslash ,$$

to ensure the required precision. A larger makes the calculation take more steps (the initial and are farther apart so it takes more steps to converge) but gives more precision. The constants and can be calculated with quickly converging series.

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 5.0 earthquake releases 32 times and a 6.0 releases 1000 times the energy of a 4.0., section 4.4. Apparent magnitude measures the brightness of stars logarithmically., section 8.3, p. 231 In chemistry the negative of the decimal logarithm, the decimal , is indicated by the letter p. For instance, pH is the decimal cologarithm of the activity of hydronium ions (the form hydrogen take in water). The activity of hydronium ions in neutral water is 10⁻⁷ mol·L⁻¹, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10⁴ of the activity, that is, vinegar's hydronium ion activity is about .

Semi-log plot (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. In such graphs, exponential functions of the form appear as straight lines with slope equal to the logarithm of . Log-log plot graphs scale both axes logarithmically, which causes functions of the form to be depicted as straight lines with slope equal to the exponent . This is applied in visualizing and analyzing ., section 34

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 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.

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. Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.

Logarithms are used for maximum-likelihood estimation of parametric statistical models. For such a model, the likelihood function depends on at least one parametric model that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the " log likelihood"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables., section 11.3

Benford's law describes the occurrence of digits in many , such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is (from 1 to 9) equals , regardless of the unit of measurement., section 2.1 Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.

The logarithm transformation is a type of data transformation used to bring the empirical distribution closer to the assumed one.

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, comparisons, where is the list's length., section 6.2.1, pp. 409–26 Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to . The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.

A function  is said to grow logarithmically if is (exactly or approximately) proportional to the logarithm of . (Biological descriptions of organism growth, however, use this term for an exponential function., chapter 19, p. 298) For example, any natural number  can be represented in binary form in no more than  . In other words, the amount of memory needed to store grows logarithmically with .

$$S\; =\; -\; k\; \backslash sum\_i\; p\_i\; \backslash ln(p\_i).\backslash ,$$

The sum is over all possible states  of the system in question, such as the positions of gas particles in a container. Moreover, is the probability that the state  is attained and is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as bits., section III.I

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 chaos theory in a deterministic way, because small measurement errors of the initial state predictably lead to largely different final states., section 1.9 At least one Lyapunov exponent of a deterministically chaotic system is positive.

$$\backslash frac\{466\}\{440\}\; \backslash approx\; \backslash frac\{493\}\{466\}\; \backslash approx\; 1.059\; \backslash approx\; \backslash sqrt122.$$

Intervals between arbitrary pitches can be measured in octaves by taking the logarithm of the frequency ratio, can be measured in equally tempered semitones by taking the logarithm ( times the logarithm), or can be measured in cents, hundredths of a semitone, by taking the logarithm ( times the logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments., chapter 5

$$\backslash frac\{x\}\{\backslash ln(x)\},$$

in the sense that the ratio of and that fraction approaches 1 when tends to infinity., theorem 4.1 As a consequence, the probability that a randomly chosen number between 1 and is prime is inversely proportional to the number of decimal digits of . A far better estimate of is given by the offset logarithmic integral function , defined by

$$\backslash mathrm\{Li\}(x)\; =\; \backslash int\_2^x\; \backslash frac1\{\backslash ln(t)\}\; \backslash ,dt.$$

The Riemann hypothesis, one of the oldest open mathematical , can be stated in terms of comparing and . The Erdős–Kac theorem describing the number of distinct also involves the natural logarithm.

The logarithm of n factorial, , is given by

$$\backslash ln\; (n!)\; =\; \backslash ln\; (1)\; +\; \backslash ln\; (2)\; +\; \backslash cdots\; +\; \backslash ln\; (n).$$

This can be used to obtain Stirling's formula, an approximation of for large ., chapter 4

$$e^a=z$$

are called complex logarithms of , when is (considered as) a complex number. A complex number is commonly represented as , where and are real numbers and is an imaginary unit, the square of which is −1. 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  by its absolute value, that is, the (positive, real) distance  to the origin, and an angle between the real () axis   and the line passing through both the origin and . This angle is called the argument of .

The absolute value of is given by

$$\backslash textstyle\; r=\backslash sqrt\{x^2+y^2\}.$$

Using the geometrical interpretation of sine and cosine and their periodicity in , any complex number  may be denoted as

 &= r (\cos \varphi + i \sin \varphi ) \\
 &= r (\cos (\varphi + 2k\pi) + i \sin (\varphi + 2k\pi)),
     

for any integer number . Evidently the argument of is not uniquely specified: both and are valid arguments of for all integers , because adding   or k⋅360° to corresponds to "winding" around the origin counter-clock-wise by  turns. The resulting complex number is always , as illustrated at the right for . One may select exactly one of the possible arguments of as the so-called principal argument, denoted , with a capital , by requiring to belong to one, conveniently selected turn, e.g. , Definition 1.6.3 or ., section 5.9 These regions, where the argument of is uniquely determined are called principal branch of the argument function.

Euler's formula connects the trigonometric functions sine and cosine to the complex exponential: $$e^\{i\backslash varphi\}\; =\; \backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi\; .$$

Using this formula, and again the periodicity, the following identities hold:, section 1.2

 &= r \left (\cos(\varphi + 2k\pi) + i \sin(\varphi + 2k\pi)\right) \\
 &= r e^{i (\varphi + 2k\pi)} \\
 &= e^{\ln(r)} e^{i (\varphi + 2k\pi)}  \\
 &= e^{\ln(r) + i(\varphi + 2k\pi)} = e^{a_k},
     

where is the unique real natural logarithm, denote the complex logarithms of , and is an arbitrary integer. Therefore, the complex logarithms of , which are all those complex values for which the  power of equals , are the infinitely many values

$$a\_k\; =\; \backslash ln\; (r)\; +\; i\; (\; \backslash varphi\; +\; 2\; k\; \backslash pi\; ),$$

for arbitrary integers .

Taking such that is within the defined interval for the principal arguments, then is called the principal value of the logarithm, denoted , again with a capital . The principal argument of any positive real number  is 0; hence is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do generalize to the principal value of the complex logarithm., theorem 6.1.

The illustration at the right depicts , confining the arguments of to the interval . This way the corresponding branch of the complex logarithm has discontinuities all along the negative real  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.e. not changing to the corresponding -value of the continuously neighboring branch. Such a locus is called a branch cut. Dropping the range restrictions on the argument makes the relations "argument of ", and consequently the "logarithm of ", multi-valued functions.

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

$$b^n\; =\; x,$$

where is an element of the group. 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 cryptography keys over unsecured information channels. Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field.

Further logarithm-like inverse functions include the double logarithm , the super-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. They are the inverse functions of the double exponential function, tetration, of , and of the logistic function, respectively., p. 357

logarithmic form appear in complex analysis and algebraic geometry as differential forms with logarithmic poles., section 2

The polylogarithm is the function defined by

$$\backslash operatorname\{Li\}\_s(z)\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \{z^k\; \backslash over\; k^s\}.$$

It is related to the natural logarithm by . Moreover, equals the Riemann zeta function .

• Decimal exponent (dex)
• Exponential function
• Index of logarithm articles

• Khan Academy: Logarithms, free online micro lectures