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In mathematics, the binary logarithm (log 2 n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x , x = log 2 n 2 x = n . {\displaystyle x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.}
Equally spaced values on a logarithmic scale have exponents that increment uniformly. Examples of equally spaced values are 10, 100, 1000, 10000, and 100000 (i.e., 10 1, 10 2, 10 3, 10 4, 10 5) and 2, 4, 8, 16, and 32 (i.e., 2 1, 2 2, 2 3, 2 4, 2 5). Exponential growth curves are often depicted on a logarithmic scale graph.
With base e the natural logarithm behaves like the common logarithm in base 10, as ln(1 e) = 0, ln(10 e) = 1, ln(100 e) = 2 and ln(1000 e) = 3 (or more precisely the representation in base e of 3, which is of course a non-terminating number).
Graphs of y = b x for various bases b: base 10, base e, base 2, base 1 / 2 . Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1 . At x = 1 , the value of y equals the base because any number raised to the power of 1 is the number itself.
For example, on a plot axis showing log 2 fold changes, an 8-fold increase will be displayed at an axis value of 3 (since 2 3 = 8). However, there is no mathematical reason to only use logarithm to base 2, and due to many discrepancies in describing the log 2 fold changes in gene/protein expression, a new term "loget" has been proposed. [10]
The logarithmic decrement can be obtained e.g. as ln(x 1 /x 3).Logarithmic decrement, , is used to find the damping ratio of an underdamped system in the time domain.. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped.
Therefore, the bel represents the logarithm of a ratio between two power quantities of 10:1, or the logarithm of a ratio between two root-power quantities of √ 10:1. [ 16 ] Two signals whose levels differ by one decibel have a power ratio of 10 1/10 , which is approximately 1.258 93 , and an amplitude (root-power quantity) ratio of 10 1/20 ...
In a third layer, the logarithms of rational numbers r = a / b are computed with ln(r) = ln(a) − ln(b), and logarithms of roots via ln n √ c = 1 / n ln(c).. The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers 2 i close to powers b j of other numbers b is comparatively easy, and series representations of ln(b) are ...