In mathematics, the logarithm is an inverse exponential function. This means that the logarithm of lg is the power to raise the number b 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 = 10 3
In this case, this is the logarithm of lg in base ten. It is equal to three.
lg 10 1000 = 3
In general, the expression will look like this:
log b x = a
The 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 determines the relationship between exponentiation and the logarithm:
log b x = a if b a = x.
History
The history of the logarithm (lg) originates in seventeenth-century Europe. This discovery of a new function has expanded 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 wonderful rules of logarithms"). Prior to the scientist’s invention, other methods existed in similar areas, such as the use of progression tables developed by Jost Burggie around 1600.
The decimal logarithm of lg is the logarithm with a base equal to ten. For the first time, valid logarithms were used with heuristic methods to convert the operation of multiplication into addition, which facilitated quick calculation. Some of these methods used tables derived from trigonometric identities.
The discovery of the function, which is now known as the logarithm (lg), is associated with an attempt to make the square of the rectangular hyperbola Gregory de Saint Vincent, a Belgian residing in Prague.
Using
Logarithms are often used outside of mathematics. Some of these cases are related to the concept of scale invariance. For example, each chamber of the nautilus shell is an approximate copy of the next, reduced or enlarged a certain number of times. This is called a logarithmic spiral.
Dimensions of self-similar geometric shapes, parts of which look similar to the final product, are also based on logarithms. Logarithmic scales are useful for quantifying the relative change in value. Moreover, since the log b x function grows very slowly at large x, logarithmic scales are used to compress large-scale scientific data. Logarithms are also found in numerous scientific formulas such as the Fenske equation or the Nernst equation.
Calculation
Some logarithms can be easily calculated, for example, log 10 1000 = 3. In the general case, they can be calculated using power series or the arithmetic mean value or extracted from a pre-calculated table of logarithms, which is highly accurate.
The iterative method for solving equations invented by Newton 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.