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The logarithm is denoted "log b x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x "). An equivalent and more succinct definition is that the function log b is the inverse function to the function x ↦ b x {\displaystyle x\mapsto b^{x}} .
The logarithm keys (log for base-10 and ln for base-e) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation. The numerical value for logarithm to the base 10 can be calculated with the following identities: [5]
The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: = + = Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition: [ 5 ] ln ( x ⋅ y ) = ln x + ln y ...
The iterated logarithm is closely related to the generalized logarithm function used in symmetric level-index arithmetic. The additive persistence of a number , the number of times someone must replace the number by the sum of its digits before reaching its digital root , is O ( log ∗ n ) {\displaystyle O(\log ^{*}n)} .
A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right).
A logarithmic unit is a unit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the ...
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The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface.