People use logarithms to make calculations easier. For instance, two numbers can be multiplied just by using a logarithm table and adding.
because
because
because
because
Cancelling exponentials
Logarithms and exponentials (antilogarithms) with the same base cancel each other.
because
because
Changing the base
This identity is needed to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log10, but not for log2. To find log2(3), you have to calculate log10(3) / log10(2) (or ln(3)/ln(2), which is the same thing).
This formula has several consequences:
Trivial identities
because
because
Calculus identities
Limits
The last limit is often summarized as "logarithms grow more slowly than any power or root of x".
Logarithms of positive numbers using the number 10 as the base are called common logarithms; those using the number e (see separate article) as the base are called natural logarithms or Napierian logarithms (for John Napier).
Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7.
This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed.
Logarithms with 10 as a base are called " common" or " Briggsian." The other exception is when the base is the number e (which equals 2.718282...).
Since tables of logarithms show positive mantissas only, a logarithm such as -5.8111 must be converted to.1889 - 6 before a table can be used to find the "antilogarithm," which is the name given to the number whose logarithm it is. A calculator will show the antilogarithm without such a conversion.
The logarithms he invented, however, were not the simple logarithms we use today (his logarithms were not what are now called "Napierian"), Shortly after Napier published his work, Briggs, an English mathematician met with him and together they worked out logarithms that much more closely resemble the common logarithms that we use today.
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