In mathematics, logarithm is the power to which a number, called a base, has to be raised to give another number—such that, in equation form, any number y can be written in the form:

whereby n is then the "logarithm" to the "base" x of the number y. [1] The use of logarithms is that they reverse the idea of a power, or exponent. Take, for instance, the formula:

which means "x is equal to 2 to the power of y". The equation can be turned around and reexpressed in logarithms as: [3]

Properties
Logarithms have the useful property that when numbers are multiplied, there logarithms are added:

Likewise, when numbers are divided, their logarithms are subtracted from one another:

Common logarithms
If the base is 10, the logarithms are called "common logarithms", which has the form:

which has common usage in many applications in science and engineering.

Natural logarithms
If the base is e (where e = 2.71828 ...) the logarithms are called either "natural logarithms" or Napierian logarithms, named after Scottish mathematician John Napier (1550-1617), the inventor of logarithms, which has the form:

or

which are taken to be equivalent notation formats. The natural logarithm has widespread in pure mathematics, especially calculus.

Notation confusion
The natural logarithm, to note, may often tend to be written with the base e "assumed" simply as:

This implied e base assumption, however, often times may become confused, often among chemists and engineers, as being a common logarithm, base 10 assumed. Chemists, for example, frequently use the symbol log(y) without a subscript for the common logarithm, i.e. base 10 assumed. [4] Hence, when a modern physical chemist reads, for example, Leo Szilard’s 1929 finding that the entropy produced by the measurement made by a Maxwell’s demon of the position of a gas particle in a two chamber system has the value:

The physical chemist may very well assume that Szilard is using a common logarithm, whereas in fact he is using a natural logarithm. [5] Subsequently, if the very same physical chemist next reads Gilbert Lewis' 1930 derivation on the same subject, finding that he calculates the entropy change to be: [6]

confusion may result on the puzzlement as to why Szilard is using common logarithms and Lewis is using natural logarithms, whereas in fact they are both using natural logarithms.

History
Logarithms were introduced by Scottish mathematician John Napier (1550-1617) in the early 17th century as a means to simplify calculations, before the advent of electronic calculators. The shorthand notation:

Log.

as an abbreviation for "logarithm" first appears in the 1616 A Description of the Admirable Table of Logarithmes, an English translation Napier's work by English mathematician Edward Wright (1561–1615). [2] The notation log (without a period, lower case "l") appears in the 1647 edition of Clavis mathematicae by English mathematician William Oughtred (1574-1660). The present-day notion of logarithms comes from Swiss mathematician Leonhard Euler (1707-1783), who connected logarithms to the exponential function:

The so-called "ln" notation, the abbreviation ln short for natural logarithm, was, supposedly, first used in 1893 by American mathematician Irving Stringham (1847-1909) in his Uniplanar Algebra.

References
1. Daintith, John. (2005). Oxford Dictionary of Science (pg. 485). Oxford University Press.
2. Cajori, Florian. (2008). “Earliest Uses of Function Symbols.”, Jeff560.tripod.com.
3. Brown, Julian. (2002). Minds, Machines, and the Quest for the Multiverse (§Appendix A: Logarithms, pg. 347). Simon and Schuster.
4. Mortimer, Robert G. (2005). Mathematics for Physical Chemistry (pg. 9). Academic Press.
5. Szilárd, Leó. (1929). “On the Decrease in Entropy in a Thermodynamic System by the Intervention of Intelligent Beings” (Uber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen), Zeitschrift fur Physik, 53, 840-56.
6. Lewis, Gilbert. (1930). “The Symmetry of Time in Physics”, Science, 71:569-77, Jun 6.