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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}} .
A graph of the common logarithm of numbers from 0.1 to 100. In mathematics, the common logarithm is the logarithm with base 10. [1] It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm.
The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see complex logarithm for more.
On the region consisting of complex numbers that are not negative real numbers or 0, the function is the analytic continuation of the natural logarithm. The values on the negative real line can be obtained as limits of values at nearby complex numbers with positive imaginary parts.
One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.
Signed zero is zero with an associated sign.In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are equivalent. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero), regarded as equal by the numerical comparison operations but with possible different behaviors in ...
Since the common logarithm of a power of 10 is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For positive numbers less than 1, the characteristic makes the resulting logarithm negative, as required. [38]
Its derivative is zero when is non-zero: () =. This follows from the differentiability of any constant function , for which the derivative is always zero on its domain of definition. The signum sgn x {\displaystyle \operatorname {sgn} x} acts as a constant function when it is restricted to the negative open region x < 0 , {\displaystyle ...