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Logarithm Online Video

Introduction to Logarithms

The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A.

Soon the new function was appreciated by Christiaan Huygens , and James Gregory. Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in that [30].

By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially astronomy.

They were critical to advances in surveying , celestial navigation , and other domains. Pierre-Simon Laplace called logarithms.

A key tool that enabled the practical use of logarithms was the table of logarithms. Briggs' first table contained the common logarithms of all integers in the range 1—, with a precision of 14 digits.

Subsequently, tables with increasing scope were written. Base logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers.

The common logarithm of x can be separated into an integer part and a fractional part , known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.

Thus using a three-digit log table, the logarithm of is approximated by. Greater accuracy can be obtained by interpolation :.

The value of 10 x can be determined by reverse look up in the same table, since the logarithm is a monotonic function. The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms.

For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis , which relies on trigonometric identities.

Calculations of powers and roots are reduced to multiplications or divisions and look-ups by. Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions.

Another critical application was the slide rule , a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule , was invented shortly after Napier's invention.

William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. 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. As illustrated above, logarithms can have a variety of bases.

A binary logarithm, or a logarithm to base 2, is applied in computing, while the field of economics utilizes base e , and in education base 10, written simply as log x, log 10 x or lg x, is used.

By organizing numbers according to these bases, real numbers can be expressed far more simply. Custom Base Logarithm: log.

Natural Logarithm Base e : ln. Base Logarithm: lg. Base-2 Logarithm: lb. In cooperation with the English mathematician Henry Briggs , Napier adjusted his logarithm into its modern form.

For the Naperian logarithm the comparison would be between points moving on a graduated straight line, the L point for the logarithm moving uniformly from minus infinity to plus infinity, the X point for the sine moving from zero to infinity at a speed proportional to its distance from zero.

Furthermore, L is zero when X is one and their speed is equal at this point. This change produced the Briggsian, or common, logarithm.

Napier died in and Briggs continued alone, publishing in a table of logarithms calculated to 14 decimal places for numbers from 1 to 20, and from 90, to , In the Dutch publisher Adriaan Vlacq brought out a place table for values from 1 to ,, adding the missing 70, values.

Both Briggs and Vlacq engaged in setting up log trigonometric tables. Such early tables were either to one-hundredth of a degree or to one minute of arc.

In the 18th century, tables were published for second intervals, which were convenient for seven-decimal-place tables. In general, finer intervals are required for calculating logarithmic functions of smaller numbers—for example, in the calculation of the functions log sin x and log tan x.

The availability of logarithms greatly influenced the form of plane and spherical trigonometry. The procedures of trigonometry were recast to produce formulas in which the operations that depend on logarithms are done all at once.

The recourse to the tables then consisted of only two steps, obtaining logarithms and, after performing computations with the logarithms, obtaining antilogarithms.

Info Print Print. Table Of Contents. Submit Feedback. Thank you for your feedback. Introduction Properties of logarithms History of logarithms.

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In cooperation with the English mathematician Henry BriggsNapier adjusted his logarithm into its modern form. The following table lists common notations for logarithms to these bases and Russisches Roulette Spielen fields where they are used. A logarithm of a real number is Treasure Hunt Cards exponent to which a base, that is, a different fixed number, needs to be increased in order to generate that real number.

This calculator can be used to determine any type of logarithm of a real number of any base you wish. Common, binary and natural logarithms can all be found using the online logarithm calculator.

A logarithm of a real number is the exponent to which a base, that is, a different fixed number, needs to be increased in order to generate that real number.

To illustrate, take the number 10, to base The logarithm of this real number will be 4. This is because 10, is equivalent to 10 to the power of 4.

Thus, just as division is the opposite mathematical operation to multiplication, the logarithm is the opposite operation to exponentiation.

Traditionally, a base of 10 is assumed in logarithms, but a base can be any number except 1. The binary logarithm of x is typically written as log 2 x or lb x.

However, a base of e is typically written as ln x and rarely as log e x. As illustrated above, logarithms can have a variety of bases.

Multiplication, the next-simplest operation, is undone by division : if you multiply x by 5 to get 5 x , you then can divide 5 x by 5 to return to the original expression x.

Logarithms also undo a fundamental arithmetic operation, exponentiation. Exponentiation is when you raise a number to a certain power.

For example, raising 2 to the power 3 equals 8 :. The general case is when you raise a number b to the power of y to get x :.

The number b is referred to as the base of this expression. It is easy to make the base the subject of the expression: all you have to do is take the y -th root of both sides.

This gives:. It is less easy to make y the subject of the expression. Logarithms allow us to do this:. This expression means that y is equal to the power that you would raise b to, to get x.

This operation undoes exponentiation because the logarithm of x tells you the exponent that the base has been raised to. This subsection contains a short overview of the exponentiation operation, which is fundamental to understanding logarithms.

Raising b to the n -th power, where n is a natural number , is done by multiplying n factors equal to b.

The n -th power of b is written b n , so that. Exponentiation may be extended to b y , where b is a positive number and the exponent y is any real number.

Finally, any irrational number a real number which is not rational y can be approximated to arbitrary precision by rational numbers.

The logarithm of a positive real number x with respect to base b [nb 1] is the exponent by which b must be raised to yield x.

In other words, the logarithm of x to base b is the solution y to the equation [5]. Several important formulas, sometimes called logarithmic identities or logarithmic laws , relate logarithms to one another.

The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms.

The logarithm of the p -th power of a number is p times the logarithm of the number itself; the logarithm of a p -th root is the logarithm of the number divided by p.

The following table lists these identities with examples. The logarithm log b x can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:.

Typical scientific calculators calculate the logarithms to bases 10 and e. Among all choices for the base, three are particularly common.

In mathematical analysis , the logarithm base e is widespread because of analytical properties explained below. On the other hand, base logarithms are easy to use for manual calculations in the decimal number system: [8].

The next integer is 4, which is the number of digits of Both the natural logarithm and the logarithm to base two are used in information theory , corresponding to the use of nats or bits as the fundamental units of information, respectively.

The following table lists common notations for logarithms to these bases and the fields where they are used. In computer science log usually refers to log 2 , and in mathematics log usually refers to log e.

The history of logarithm in seventeenth-century Europe is the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods.

The common logarithm of a number is the index of that power of ten which equals the number. Some of these methods used tables derived from trigonometric identities.

Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A.

Soon the new function was appreciated by Christiaan Huygens , and James Gregory. Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in that [30].

By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially astronomy.

They were critical to advances in surveying , celestial navigation , and other domains. Pierre-Simon Laplace called logarithms. A key tool that enabled the practical use of logarithms was the table of logarithms.

Briggs' first table contained the common logarithms of all integers in the range 1—, with a precision of 14 digits. Subsequently, tables with increasing scope were written.

Base logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers.

The common logarithm of x can be separated into an integer part and a fractional part , known as the characteristic and mantissa.

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

Thus using a three-digit log table, the logarithm of is approximated by. Greater accuracy can be obtained by interpolation :.

The value of 10 x can be determined by reverse look up in the same table, since the logarithm is a monotonic function. The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms.

For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis , which relies on trigonometric identities.

Calculations of powers and roots are reduced to multiplications or divisions and look-ups by. Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions.

Another critical application was the slide rule , a pair of logarithmically divided scales used for calculation.

The non-sliding logarithmic scale, Gunter's rule , was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other.

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.