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Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at =, where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis ( 1 , ∞ ) {\displaystyle (1,\infty )} .
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.
The mathematical notation for using the common logarithm is log(x), [4] log 10 (x), [5] or sometimes Log(x) with a capital L; [a] on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log".
The graph of the logarithm base 2 crosses the x-axis at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log 2 (8) = 3 and 2 3 = 8. The graph gets arbitrarily close to the y-axis, but does not meet it. Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations.
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4. The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1. The minimum value of x is ...
For example, setting 7.5 on one scale over 10 on the other scale, the user can see that at the same time 1.5 is over 2, 2.25 is over 3, 3 is over 4, 3.75 is over 6, 4.5 is over 6, and 6 is over 8, among other pairs.
A simple arithmetic calculator was first included with Windows 1.0. [5]In Windows 3.0, a scientific mode was added, which included exponents and roots, logarithms, factorial-based functions, trigonometry (supports radian, degree and gradians angles), base conversions (2, 8, 10, 16), logic operations, statistical functions such as single variable statistics and linear regression.
Mathematical tables are lists of numbers showing the results of a calculation with varying arguments.Trigonometric tables were used in ancient Greece and India for applications to astronomy and celestial navigation, and continued to be widely used until electronic calculators became cheap and plentiful in the 1970s, in order to simplify and drastically speed up computation.