บอร์ด นี้ มีสาระ: Logarithm

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10³ = 10 × 10 × 10. More generally, if x = b^y, then y is the logarithm of x to base b, and is written log_b(x), so log₁₀(1000) = 3.

Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by scientists, engineers, and others to perform computations more easily and rapidly, using slide rules and logarithm tables. These devices rely on the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

$\log_b(xy) = \log_b (x) + \log_b (y). \,$

The present-day notion of logarithms comes from Leonard Euler, who connected them to the exponential function in the 18th century.

The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant e(≈ 2.718) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base b = 2 and is prominent in computer science.

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a logarithmic unit quantifying sound pressure and voltage ratios. In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulas, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.

In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant; it has applications in public-key cryptography.

Motivation and definition

The idea of logarithms is to reverse the operation of exponentiation, that is raising a number to a power. For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2:

$2^3 = 2 \times 2 \times 2 = 8. \,$

It follows that the logarithm of 8 with respect to base 2 is 3.

Exponentiation

The third power of some number b is the product of 3 factors of b. More generally, raising b to the n-th power, where n is a natural number, is done by multiplying n factors. The n-th power of b is written bⁿ, so that

$b^n = \underbrace{b \times b \times \cdots \times b}_{n \text{ factors}}.$

The n-th power of b, bⁿ, is defined whenever b is a positive number and n is a real number. For example, b⁻¹ is the reciprocal of b, that is, 1/b.^{[nb 1]}

Definition

The logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. In other words, the logarithm of x to base b is the solution y of the equation^[2]

$b^y = x. \,$

The logarithm is denoted "log_b(x)" (pronounced as "the logarithm of x to base b" or "the "base-b logarithm of x"). In the equation y = log_b(x), the value y, is the answer to the question "To what power must b be raised, in order to yield x?". For the logarithm to be defined, the base b must be a positive real number not equal to 1 and x must be a positive number.^{[nb 2]}

Examples

For example, log₂(16) = 4, since 2⁴ = 2 ×2 × 2 × 2 = 16. Logarithms can also be negative:

$\log_2 \!\left( \frac{1}{2} \right) = -1,\,$

since

$2^{-1} = \frac 1 {2^1} = \frac 1 2.$

A third example: log₁₀(150) is approximately 2.176, which lies between 2 and 3, just as 150 lies between 10² = 100 and 10³ = 1000. Finally, for any base b, log_b(b) = 1 and log_b(1) = 0, since b¹ = b and b⁰ = 1, respectively.

Logarithmic identities

Several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another.^[3]

Product, quotient, power, and root

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. Therefore, 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:

	Formula	Example
product	$\log_b(x y) = \log_b (x) + \log_b (y) \,$	$\log_3 (243) = \log_3(9 \cdot 27) = \log_3 (9) + \log_3 (27) = 2 + 3 = 5 \,$
quotient	$\log_b \!\left(\frac x y \right) = \log_b (x) - \log_b (y) \,$	$\log_2 (16) = \log_2 \!\left ( \frac{64}{4} \right ) = \log_2 (64) - \log_2 (4) = 6 - 2 = 4$
power	$\log_b(x^p) = p \log_b (x) \,$	$\log_2 (64) = \log_2 (2^6) = 6 \log_2 (2) = 6 \,$
root	$\log_b \sqrt[p]{x} = \frac {\log_b (x)} p \,$	$\log_{10} \sqrt{1000} = \frac{1}{2}\log_{10} 1000 = \frac{3}{2} = 1.5$

Change of base

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:

$\log_b(x) = \frac{\log_k(x)}{\log_k(b)}.\,$

Typical scientific calculators calculate the logarithms to bases 10 and e.^[4] Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:

$\log_b (x) = \frac{\log_{10} (x)}{\log_{10} (b)} = \frac{\log_{e} (x)}{\log_{e} (b)}. \,$

Given a number x and its logarithm log_b(x) to an unknown base b, the base is given by:

$b = x^\frac{1}{\log_b(x)}.$

Particular bases

Among all choices for the base b, three are particularly common. These are b = 10, b = e (the irrational mathematical constant ≈ 2.71828), and b = 2. In mathematical analysis, the logarithm to base e is widespread because of its particular analytical properties explained below. On the other hand, base-10 logarithms are easy to use for manual calculations in the decimal number system:^[5]

$\log_{10}(10 x) = \log_{10}(10) + \log_{10}(x) = 1 + \log_{10}(x).\$

Thus, log₁₀(x) is related to the number of decimal digits of a positive integer x: the number of digits is the smallest integer strictly bigger than log₁₀(x).^[6] For example, log₁₀(1430) is approximately 3.15. The next integer is 4, which is the number of digits of 1430. The logarithm to base two is used in computer science, where the binary system is ubiquitous.

The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write log(x) instead of log_b(x), when the intended base can be determined from the context. The notation ^blog(x) also occurs.^[7] The "ISO notation" column lists designations suggested by the International Organization for Standardization (ISO 31-11).^[8]

Base b	Name for log_b(x)	ISO notation	Other notations	Used in
2	binary logarithm	lb(x)^[9]	ld(x), log(x) (in computer science), lg(x)	computer science, information theory
e	natural logarithm	ln(x)^{[nb 3]}	log(x) (in mathematics and many programming languages^{[nb 4]})	mathematical analysis, physics, chemistry, statistics, economics, and some engineering fields
10	common logarithm	lg(x)	log(x) (in engineering, biology, astronomy),	various engineering fields (see decibel and see below), logarithm tables, handheld calculators

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Saturday, October 22, 2011

Logarithm