Many of us students have difficult time with logarithms. The problem can be that we memories the rules without fully understanding them. And many of us thinks; why waste our time on these archaic entities; we are never going to see them again anyway. But logarithms are useful in many fields from finance to astronomy.

There are three sorts of logarithms,

• Binary logarithms (base 2 – symbol: lb x)
• Natural logarithms (base e – symbol: ln x)
• Common logarithms (base 10 – symbol: lg or log)

We are in this project just going to tell you some things about the common logarithms. This logarithms are also often called Briggs’s logarithms, you will know why later in this text.

Logarithms are numbers that are known in algebra as exponents.

Exponents are used to express repeated multiplications of a single number. Multiplications is a shortcut for addition.

Exponents are a shortcut for multiplication.

You can say that logarithm is a shortcut for exponents.

• Example:

X = aÙ log aÙ x

This means, a logarithm is an exponent which defines to what power the base a must be raised in order to give x, called the antilogarithm. Only positive real numbers are acceptable as bases for workable systems of logarithms.

Logarithms are important for theoretical purposes. They were invented for about 400 years ago. Before that, mathematician, astronomers and so on, had to use a lot of time to simple mathematics such as multiplication or division.

Then logarithms were invented and they made big multiplicatoions and divisions much more simple. At first you had to use a table when you were going to use logarithms, but for about 20 years ago the calculator with a log-function came and that made it even more simple.

There are three men that are important for the inventing of the logs, we will now tell you who, and a little bit about them.

• John Napier

John Napier was the inventor of logarithms. He was born in Merchiston Castle in Edinburg in 1550.He educated at the university of St.Andrews ,and he entering the university in 1563 . The common and natural systems of logarithms used today, do not employ the same base as Napiers logarithms, although natural logarithms are sometimes called Naperian logarithms.

As base for his powers he decided on

(1 - 10Ù -7)

By multiplying all powers by

10Ù 7,

we have the relation

N = 10Ù 7 (1-10Ù -7) Ù L

Where L is the "Naperian logarithm" of the number N.

Napiers logarithms are not really to any base although in our present terminology, it is unreasonable (but perhaps a little misleading) to say that they are to base 1/e. And e has here something to do with natural logarithms and we are not going to go any deeper into them. Certainly they involve a construction in a way that we will now explain:

"Napier did not think of logarithms in a algebraic way, in fact algebra was not well enough developed in Napiers time to make this realistic approach. Rather he thought by dynamical analogy."

The Swuss watchmaker Joost Bürgi, maker of astronomical instruments and an indefatigable computer and assistant to Johannes Kepler, Imperial astronomer in Prague, also conceived a system of logarithms to facilitate the mutiplicastion of large numbers.Where Napier decided on powers of ( 1-10^7 ),Bürgi made the better choice,

(1+10^-4),

this made his power indices increase as the power numbers increased, while Napier`s decreased.

There was another difference between the work of the two men: where Napier multiplied his power by 10^7, Bürgi chose

10^8

he also multiplied his logarithms by 10 in his tables.

N= 10^8(1+10-^4)^L.

Bürgi called 10L the "red numbers" corresponding to the " black number" N. If all black numbers are divided by 10^4, we obtain what is nearly a system of logarithms to the base e. Which also ha something to do with natural logarithms.

.

The matter of who was "first" must remain unsettled but the official priority belongs to Napier because of his publishing date, 1614, six years before Bürgi. It may be that he had began his work in 1588, or even 1584. Napier is reported to have discussed his pwn results with Brahe in 1594.

So all we know is that they did this work independent from each other.

Henry Briggs was born in Yorkshire, England. His occupation is unknown, but they were probably poor. Witch indicates that he could not has attended Cambridge without financial assistance from his college. He was educated at Cambridge University from 1577 to 1585.

In 1592 he became reader of the Linacre lecture and in 1619 he was appointed professor of geometry at Oxford. Briggs published works on navigation, astronomy, and mathematics. Virtually the whole of his work in mathematics was devoted to making computation more easy. Briggs held the position as Professor of Geometry at Oxford in the last decade of his life, until he died there in Oxford.

Henry Briggs was the man most responsible for scientists acceptance of logarithms. In his lectures at Gresham he proposed that Napier’s logarithms would be more useful if they were to base 10, so called "common" logarithms. After travelling to Edinburgh on two occasions to visit Napier, he constructed a table of logarithms in 1617, that was used until the 19^th century. Also called Briggsian logarithms.

His writings were mainly responsible for the widespread acceptance of logarithms throughout Europe. This is the logarithms you get if you use the log-function on your calculator.

If you are going to write a number as a potens of 10, you must use the log-function on the calculator

The number you must upturn 10 in to get p , we call the logarithm to p.

• You have 2 parents, so you have 2 ancestors in the first generation. This calculation can be expressed as

2Ù 1 = 2.

• Each of your parents has two parents, so in the second generation you have

2 × 2 = 2Ù 2 = 4 ancestors

• Each of your grandparents has 2 parents, so in the third generation you have

4 × 2 = 2 × 2 × 2 = 2 Ù 3 = 8 ancestors

The calculation continues in this pattern.

That is when exponent x is true that

2Ù x = 1024.

We will now tell you how they had to do it before logarithms were invented, after, and by using the log-function on the calculator.

• Before

You can find the answer by multiplying 2 by itself until you reach 1024.

2 × 2 × 2……

keep on trying till you reach 1024! That could take a really long time, of course you can be lucky and find it after four tries or less.

But to be lucky once in a wile is not the way to be a good mathematician!

• After

In many years after the logarithms were invented they had to use the briggsian table. We have copied the piece of the table that they would need. And we will now show you how they used it.

• With the log-function

To simplify it even more we now just use the log-function on the calculator and if you can all the rules and understands them, logarithms now shouldn’t be too difficult…

2Ù x = 1024

log 2Ù x = log 1024

x × log 2 = log 1024

x = log 1024 / log 2 = 3,01 / 0.3 = 10

With not much problems you know the answer is 10.

John Napier, Henry Briggs, and Joost Bürgi invented a good system that many people before us, and sure many people after us will have a lot use for. I think many people are very thankful for what he did to the mathematics. Now you can find the correct answer to many questions, while before Briggs fantastic system, they had to use a lot more time to find the correct answer. Even if many student inclusive us have hard time with logarithms, we probably would have had bigger problems without them!