लघुगणक

विकिपीडिया, एक मुक्त ज्ञानकोष से

अलग-अलग आधार के लिये लघुगणकीय फलन का आरेखण: लाल रंग वाला e, हरा रंग वाला 10, तथाबैगनी वाला 1.7. सभी आधारों के लघुगणक बिन्दु (1, 0) से होकर गुजरते हैं क्योंकि किसी भी संख्या पर शून्य घातांक का मान 1 होता है।

गणित में किसी दिए हुए आधार पर किसी संख्या का लघुगणक वह संख्या होती है जिसको उस आधार के उपर घात लगाने सेउसका मान दी हुई संख्या के बराबर हो जाय। उदाहरण के लिये , १० आधार पर १००००० (एक लाख) का लघुगणक ५ होगा क्योंकि आधार १० पर ५ घात लगाने से उसका मान १००००० हो जाता है।

अर्थात किसी संख्या x, आधार b और घातांक n, के लिये

$\mbox{if}~~ b^n = x, ~~\mbox{then}~~ \log_b (x) = n. \,$

[संपादित करें] लघुगणकों के गुण

जब x और b दोनो धनात्मक वास्तविक संख्याएँ हों तो, log_b(x) का मान एक अद्वितीय वास्तविक संख्या होती है। The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers. ^[१]^[२]

The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:

$b^x \times b^y = b^{x+y} \ ,$

which by taking logarithms becomes

$\log_b \left(b^x \times b^y \right) = \log_b \left( b^{x+y} \right)$ $\ = x + y = \log_b \left(b^x \right) + \log_b \left(b^y \right). \$

A related property is reduction of exponentiation to multiplication. Using the identity:

$c = b^{\log_b (c )} \ ,$

it follows that c to the power p (exponentiation) is:

$c^p = \left(b^{\log_b (c )}\right)^p = b^{p \log_b (c )} \ ,$

or, taking logarithms:

$\log_b \left(c^p \right) = p \log_b (c ) \ .$

In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = b^product.

Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. Logarithms make lengthy numerical operations easier to perform. The whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. Although the above practical advantages are not important for numerical work today, they are used in graphical analysis (see Bode plot).

[संपादित करें] लघुगणकों के उपयोग

Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely related functions. In the equation bⁿ = x, b can be determined with radicals, n with logarithms, and x with exponentials. See logarithmic identities for several rules governing the logarithm functions.

[संपादित करें] विज्ञान

Various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a more complete list.

In chemistry, the negative of the base-10 logarithm of the concentration of hydronium ions (H₃O⁺, the form H⁺ takes in water) is the measure known as pH. The concentration of hydronium ions in neutral water is 10⁻⁷ mol/L at 25 °C, hence a pH of 7.

The bel (symbol B) is a unit of measure which is the base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. The Bel is named after telecommunications pioneer Alexander Graham Bell. The decibel (dB), equal to 0.1 bel, is more commonly used. The neper is a similar unit which uses the natural logarithm of a ratio.

The Richter scale measures earthquake intensity on a base-10 logarithmic scale.

In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B.

In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.

In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation.

In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using comparison can require time proportional to the product N × log N. Similarly, base-2 logarithms are used to express the amount of storage space or memory required for a binary representation of a number—with k bits (each a 0 or a 1) one can represent 2^k distinct values, so any natural number N can be represented in no more than (log₂ N) + 1 bits.

Similarly, in information theory logarithms are used as a measure of 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 log₂ N bits.

In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.

Many types of engineering and scientific data are typically graphed on log-log or semilog axes, in order to most clearly show the form of the data.

Musical intervals are measured logarithmically as semitones. The interval between two notes in semitones is the base-2^1/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Fractional semitones are used for non-equal temperaments. Especially to measure deviations from the equal tempered scale, intervals are also expressed in cents (hundredths of an equally-tempered semitone). The interval between two notes in cents is the base-2^1/1200 logarithm of the frequency ratio (or 1200 times the base-2 logarithm). In MIDI, notes are numbered on the semitone scale (logarithmic absolute nominal pitch with middle C at 60). For microtuning to other tuning systems, a logarithmic scale is defined filling in the ranges between the semitones of the equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. (see microtuning in MIDI).

[संपादित करें] वाह्य सूत्र

[[en:Logarithms]

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लघुगणक

विकिपीडिया, एक मुक्त ज्ञानकोष से

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