Short answers, pulled from the story.
John Napier introduced logarithms in 1614 in a book titled Mirifici Logarithmorum Canonis Descriptio, the Description of the Wonderful Canon of Logarithms. He coined the term logarithmus from the Greek words logos and arithmos.
A logarithm of a number is the exponent to which a fixed base must be raised to produce that number. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.
The three common bases are 10, e, and 2. Base 10 is the common or decimal logarithm used in science and engineering, base e is the natural logarithm widespread in mathematics and physics, and base 2 is the binary logarithm used in computer science, information theory, music theory, and photography.
Logarithms turned multiplication and division into addition and subtraction, because the logarithm of a product is the sum of the logarithms of its factors. Using logarithm tables and slide rules, this replaced tedious multi-digit multiplication with table look-ups and simpler addition, greatly aiding astronomy, surveying, and navigation.
Henry Briggs compiled the first table of logarithms in 1617, just after Napier's invention, using 10 as the base. His first table contained the common logarithms of all integers from 1 to 1000 with a precision of 14 digits.
Earthquake strength is measured by the common logarithm of the energy released, used in the moment magnitude and Richter scales, so a 6.0 quake releases 1000 times the energy of a 4.0. In chemistry, pH is the negative decimal logarithm of hydronium ion activity, so neutral water has a pH of 7 and vinegar about 3.
The natural logarithm uses the constant e as its base and can be defined as the area under the curve of one over t. It is called natural because its derivative is the very simple expression one over x, which also explains the importance of the constant e.