Logarithm

he graph of the logarithm to base 2 crosses the x-axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log₂(8) = 3, because 2³ = 8. The graph gets arbitrarily close to the y-axis, but does not hit it.

The logarithm of a number is the exponent to which a fixed number, 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 generallyT, 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 scientific computations using slide rules and logarithm tables. These devices rely on the fact that the logarithm of a product is the sum of the logarithms of the factors:

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 engineering. The base of the natural logarithm is the constant e (≈ 2.718). It 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, measure the complexity of algorithms and of geometric objects called fractals, and appear in formulas counting prime numbers. They describe musical intervals, 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.

Chaudhry Ahmed Khan Advocate High Courts Cell: 0300 2172379