Talking about logarithms we can say that the logarithm is perhaps the single, most useful arithmetic concept in all the sciences. Thus, an understanding of them is so essential to an understanding of many scientific ideas. Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss. The logarithms which they invented differed from each other and from the common and natural logarithms now in use. Napier’s logarithms were published in 1614; Burgi’s logarithms were published in 1620. The objective of both men was to simplify mathematical calculations. Napier’s approach was algebraic and Burgi’s approach was geometric. Neither men had a concept of a logarithmic base. Napier defined logarithms as a ratio of two distances in a geometric form, as opposed to the current definition of logarithms as exponents. The possibility of defining logarithms as exponents was recognized by John Wallis in 1685 and by Johann Bernoulli in 1694.

The invention of the common system of logarithms is due to the combined effort of Napier and Henry Biggs in 1624. Natural logarithms first arose as more or less accidental variations of Napier’s original logarithms. Their real significance was not recognized until later. The earliest natural logarithms occur in 1618.

Logarithms are useful in many fields from finance to astronomy. In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number. For example, the logarithm of 1000 to the base 10 is 3, because 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32. In the language of exponents: 10^{3} = 1000, so log_{10}1000 = 3, and 2^{5} = 32, so log_{2}32 = 5. The logarithm of *x* to the base *b* is written log_{b}(*x*) or, if the base is implicit, as log(*x*). So, for a number *x*, a base *b* and an exponent *y*,

Logarithms may be defined and introduced in several different ways. But for our purposes, let’s adopt a simple approach. This approach originally arose out of a desire to simplify multiplication and division to the level of addition and subtraction. Of course, in this era of the cheap hand calculator, this is not necessary anymore but it still serves as a useful way to introduce logarithms. The question is, therefore:

Is there any operation in mathematics which produces a multiplication by the performance of an addition?

With not too much thought, the answer should come to you.

What is 2^{3} x 2^{4}.

The answer is 2 ^{7} which is obtained by adding the powers 3 and 4. This is correct, of course, since 2^{3} x 2^{4} is just seven 2s multiplied together. Note that this addition trick does not work for the case of 3^{3} x 2^{4}. The base numbers must be the same, as in the first case, where we used 2.

In general, this addition trick can be written as p^{a} x p^{b} = p^{a+b}. This expression will do our job of multiplying any two numbers, say 1.3 and 6.9, if we can only express 1.3 as p^{a} and 6.9 as p^{b}.

What number will we use for the base p? Any number will do, but traditionally, only two are in common use:

Ten (10) and the transcendental number e (= 2.71828…), giving logarithms to the base 10 or common logarithms (log), and logarithms to the base e or natural logarithms (ln).

Let’s first talk about logarithms to the base 10 or common logs. We thus choose to let our number 1.3 be equal to 10^{a}.

1.3 = 10^{a}`a’ is called “the logarithm of 1.3”. How large is `a’? Well, it’s not 0 since 10^{0} = 1 and it’s less than 1 since 10^{1} = 10. Therefore, we see that all numbers between 1 and 10 have logarithms between 0 and 1. If you look at the table below you’ll see a summary of this.

Number range 1 – 10 or 10 ^{0} – 10^{1}10 – 100 or 10 ^{1} – 10^{2}100 -1000 or 10 ^{2} – 10^{3}etc. |
Logarithm Range 0 -1 ^{ }1 – 2 ^{ }2 – 3 ^{ }etc. |

You see, we have the number range listed on the left and the logarithm range listed on the right. For numbers between 1 and 10, that is between 10^{0} and 10^{1}, the logarithm lies in the range 0 to 1. For numbers between 10 and 100, that is between 10^{1} and 10^{2}, the logarithm lies in the range 1 to 2, and so on. Now in the bad old days before calculators, you would have to learn to use a set of logarithm tables to find the logarithm of our number, 1.3, that we asked for earlier. As you can guess, today is easier to deal with logarithms due to the technological advantage of calculators and computers. (Nice advantage!).