# 1231231 - 5 Exponential and Logarithmic Functions 5.1...

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5.1 Exponential functions An exponential (or power) function is of the form a is known as the base x is known as the exponent, power or index. Remember the following rules for indices: 1. 2. 3. 4. 5. 6. a 0 1 q 2 a p a p q 1 a p a p 1 a p 2 q a pq a p a q a p q a p a q a p q 1 a 1 2 . y a x . 5 Exponential and Logarithmic Functions 109 108 5 Exponential and Logarithmic Functions 1 In this chapter we will meet logarithms, which have many important applications, particularly in the field of natural science. Logarithms were invented by John Napier as an aid to computation in the 16th century. John Napier was born in Edinburgh, Scotland, in 1550. Few records exist about John Napier’s early life, but it is known that he was educated at St Andrews University, beginning in 1563 at the age of 13. However, it appears that he did not graduate from the university as his name does not appear on any subsequent pass lists.The assumption is that Napier left to study in Europe.There is no record of where he went, but the University of Paris is likely, and there are also indications that he spent time in Italy and the Netherlands. While at St.Andrews University, Napier became very interested in theology and he took part in the religious controversies of the time. He was a devout Protestant, and his most important work, the Plaine Discovery of the Whole Revelation of St.John was published in 1593. It is not clear where Napier learned mathematics, but it remained a hobby of his, with him saying that he often found it hard to find the time to work on it alongside his work on theology. He is best remembered for his invention of logarithms, which were used by Kepler, whose work was the basis for Newton’s theory of gravitation. However his mathematics went beyond this and he also worked on exponential expressions for trigonometric functions, the decimal notation for fractions, a mnemonic for formulae used in solving spherical triangles, and “Napier’s analogies”, two formulae used in solving spherical triangles. He was also the inventor of “Napier’s bones”, used for mechanically multiplying, dividing and taking square and cube roots. Napier also found exponential epressions for trignometric functions, and introduced the decimal notation for fractions. We can still sympathize with his sentiments today, when in the preface to the Mirifici logarithmorum canonis descriptio , Napier says he hopes that his “logarithms will save calculators much time and free them from the slippery errors of calculations”. John Napier Example Simplify x 1 5 x 2 5 5 2 x 3 x 3 5 x 3 5 x 0 1 x 1 5 x 2 5 5 2 x 3 . Example Evaluate without a calculator. 8 2 3 1 3 2 8 2 1 2 2 1 4 8 2 3 Graphing exponential functions Consider the function y 2 x . x 0 1 2 3 4 5 y 1 2 4 8 16 32 1 2 1 4 1 2 The y -values double for every integral increase of x .
Now All exponential graphs of the form can be expressed in this way, and from our knowledge of transformations of functions this is actually a reflection in the y -axis.