Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the 'logarithm to base 10' of 1000 is 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x—if no confusion is possible. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so logb (x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:Starting from the defining identity In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the 'logarithm to base 10' of 1000 is 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x—if no confusion is possible. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so logb (x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is: For example, log2 64 = 6, as 26 = 64. The logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer science. Logarithms are examples of concave functions. Logarithms were introduced by John Napier in 1614 as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors: provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision.The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help describing frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, 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 uses in public-key cryptography. Addition, multiplication, and exponentiation are three fundamental arithmetic operations (or in some context the first three hyperoperations). Addition, the simplest of these, can be undone by subtraction: for example, the addition of 2 in 3 + 2 = 5 can be undone by subtracting 2: 5 − 2 = 3. Multiplication, the next-simplest operation, can be undone by division: doubling a number x, i.e., multiplying x by 2, the result is 2x. To get back x, it is necessary to divide by 2. For example, 3 × 2 = 6 and the process of multiplying by 2 is undone by dividing by 2: 6 ÷ 2 = 3. The idea and purpose of logarithms is also to undo a fundamental arithmetic operation, namely raising a number to a certain power (an operation also known as exponentiation). For example, raising 2 to the power 3 yields 8, because 8 is the product of three factors of 2: The logarithm (with respect to base 2) of 8 is 3, reflecting the fact that 2 was raised to the power 3 to get 8.