Explain Logarithm In detail
Answers
Answer:
In mathematics the logarithms is the 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 base 10" of 1000 is 3, or log10(1000) = 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, when no confusion is possible, or when the base does not matter such as in big O notation.
More generally, exponentiation allows any positive real number as base 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:
{\displaystyle \log _{b}(x)=y\ }{\displaystyle \log _{b}(x)=y\ } exactly if {\displaystyle \ b^{y}=x\ }{\displaystyle \ b^{y}=x\ } and {\displaystyle \ x>0}{\displaystyle \ x>0} and {\displaystyle \ b>0}{\displaystyle \ b>0} and {\displaystyle \ b\neq 1}{\displaystyle \ b\neq 1}.
For example, log2 64 = 6, as 26 = 64.
The logarithm 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.[1]
Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations.[2] 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:
{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,\,}{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,\,}
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.[3]
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 to describe 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 modular discrete logarithm is another variant; it has uses in public-key cryptography.
Answer---↷
✭ A logarithm is the power to which a number must be raised in order to get some other number (see Section 3 of this Math Review for more about exponents). For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100 :
→ log 100 = 2
◎ because
→ 102 =100
★ This is an example of a base-ten logarithm. We call it a base ten logarithm because ten is the number that is raised to a power. The base unit is the number being raised to a power. There are logarithms using different base units. If you wanted, you could use two as a base unit. For instance, the base two logarithm of eight is three, because two raised to the power of three equals eight:
→ log2 8 = 3
◎ because
→ 23 = 8
★ In general, you write log followed by the base number as a subscript. The most common logarithms are base 10 logarithms and natural logarithms; they have special notations. A base ten log is written
◎ log,
◎ and a base ten logarithmic equation is usually written in the form:
→ log a = r
◎ A natural logarithm is written
◎ ln,
◎ and a natural logarithmic equation is usually written in the form:
→ ln a = r
★ So, when you see log by itself, it
means base ten log. When you see ln, it
mean natural logarithm (we'll define
natural logarithms below). In this course
only base ten and natural logarithms will
be used.