5.1Exponential functionsAn exponential (or power) function is of the form ais known as the base xis known as the exponent, power or index.Remember the following rules for indices:1. 2. 3. 4. 5. 6. a01q2apapq1apap1ap2qapqapaqapqapaqapq1a12.yax.5 Exponential and Logarithmic Functions1091085 Exponential and LogarithmicFunctions1In this chapter we will meet logarithms,which have many important applications,particularly in the field of natural science.Logarithms were invented by John Napieras an aid to computation in the 16thcentury.John Napier was born in Edinburgh,Scotland, in 1550. Few records exist aboutJohn Napier’s early life, but it is known thathe was educated at St Andrews University,beginning in 1563 at the age of 13.However, it appears that he did not graduatefrom the university as his name does notappear on any subsequent pass lists.Theassumption is that Napier left to study inEurope.There is no record of where hewent, but the University of Paris is likely,and there are also indications that he spenttime in Italy and the Netherlands.While at St.Andrews University, Napier became very interested in theology and hetook part in the religious controversies of the time. He was a devout Protestant, andhis most important work, the Plaine Discovery of the Whole Revelation of St.John was publishedin 1593.It is not clear where Napier learned mathematics, but it remained a hobby of his, withhim saying that he often found it hard to find the time to work on it alongside hiswork on theology. He is best remembered for his invention of logarithms, whichwere 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 exponentialexpressions for trigonometric functions, the decimal notation for fractions, amnemonic 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 andcube roots. Napier also found exponential epressions for trignometric functions, andintroduced the decimal notation for fractions.We can still sympathize with his sentiments today, when in the preface to the Mirificilogarithmorum canonis descriptio, Napier says he hopes that his “logarithms will savecalculators much time and free them from the slippery errorsof calculations”.John NapierExampleSimplify x15x2552x3x35x35x01x15x2552x3.ExampleEvaluate without a calculator.8231328212214823Graphing exponential functionsConsider the function y2x.x012345y12481632121412The y-values double forevery integral increase of x.
Now All exponential graphs of the form canbe expressed in this way, and from our knowledge of transformations of functions this is actually a reflection in the y-axis.