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Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d.
Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in information theory. In computer science, they count the number of steps needed for binary search and related algorithms.
The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. A can be calculated using the exponential series, which converges quickly provided y is not too large. Calculating the logarithm of larger z can be reduced to smaller values of z by writing z = a · 10 b, so that ln(z) = ln(a) + b ...
The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a [4] (with the area being negative when 0 < a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural".
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).
An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the mantissa. [b] Thus, log tables need only show the fractional part. Tables of common logarithms ...
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