In mathematics, the logarithm is the inverse of the exponential function. This means that the logarithm of lg is the power to which the number b must be raised in order to get x as a result. In the simplest case, it takes into account the repeated multiplication of the same value.
Consider a specific example:
1000=10 × 10 × 10=103
In this case, it is the base ten logarithm of lg. It is equal to three.
lg101000=3
In general, the expression will look like this:
lgbx=a
Exponentiation allows any positive real number to be increased to any real value. The result will always be greater than zero. Therefore, the logarithm for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number a. Moreover, it defines the relation between exponentiation and logarithm:
lgbx=a if ba=x.
History
The history of the logarithm (lg) originates in Europe in the seventeenth century. This is the opening of a new featureexpanded the scope of analysis beyond algebraic methods. The method of logarithms was publicly proposed by John Napier in 1614 in a book called Mirifici Logarithmorum Canonis Descriptio ("Description of the Remarkable Rules of Logarithms"). Prior to the invention of the scientist, there were other methods in similar areas, such as the use of progression tables developed by Jost Bürggi around 1600.
The decimal logarithm lg is the logarithm with base ten. For the first time, real logarithms were used with heuristics to convert multiplication to addition, facilitating fast computation. Some of these methods used tables derived from trigonometric identities.
The discovery of the function now known as the logarithm (lg) is attributed to Gregory de Saint Vincent, a Belgian living in Prague, attempting to quadrature a rectangular hyperbola.
Use
Logarithms are often used outside of mathematics. Some of these cases are related to the notion of scale invariance. For example, each chamber of the nautilus shell is an approximate copy of the next, reduced or enlarged by a certain number of times. This is called a logarithmic spiral.
Dimensions of self-made geometries, parts of which look similar to the final product, are also based on logarithms. Logarithmic scales are useful for quantifying relative changevalues. Moreover, since the function logbx grows very slowly at large x, logarithmic scales are used to compress large-scale scientific data. Logarithms also appear in numerous scientific formulas such as the Fenske equation or the Nernst equation.
Calculation
Some logarithms can be easily calculated, for example log101000=3. In general, they can be calculated using power series or the arithmetic-geometric mean, or extracted from a pre-calculated table logarithms, which has high accuracy.
Newton's iterative method for solving equations can also be used to find the value of the logarithm. Since the inverse function for the logarithmic is exponential, the calculation process is greatly simplified.
