# The natural exponential function ln x is a one to one

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The Natural Exponential Functionlnxis a one-to-one function, since it is increasing; therefore it hasan inverse which we shall call exp(x).The natural base is thenumberesuch thatlne= 1exp(1) =e.The value ofecan be found using ln 1 = 0 and ln (1) = 1:1 = limh0ln(1 +h)-ln 1h= limh0ln(1+h)1/h= lnlimh0(1 +h)1/h.Thuse= limh0(1 +h)1/h2.7182818284590452354. . .Chapter 6 Lecture NotesNine LecturesMAT186H1F Lec0107 BurbullaChapter 6: Applications of Integration6.1 Velocity and Net Change6.2 Regions Between Curves6.3 Volume by Slicing6.4 Volume by Shells6.5 Length of Curves6.6 Surface Area6.7 Physical Applications6.8 Logarithmic and Exponential Functions Revisited6.9 Exponential Models6.10 Hyperbolic Functions: An OverviewProperties of the Natural Exponential Function1.The domain of exp(x) isR2.The range of exp(x) isy>03.ln(exp(x)) =xfor allx4.exp(lnx) =xforx>05.exp (x) = exp(x)6.exp(x+y) = exp(x) exp(y)7.exp(x-y) = exp(x)/exp(y)8.(exp(x))y= exp(xy)Note: we usually write exp(x) =ex,whereeis the natural base.Chapter 6 Lecture NotesNine LecturesMAT186H1F Lec0107 Burbulla
Chapter 6: Applications of Integration6.1 Velocity and Net Change6.2 Regions Between Curves6.3 Volume by Slicing6.4 Volume by Shells6.5 Length of Curves6.6 Surface Area6.7 Physical Applications6.8 Logarithmic and Exponential Functions Revisited6.9 Exponential Models6.10 Hyperbolic Functions: An OverviewSome ProofsProperties 1, 2, 3 and 4 follow directly from the fact that exp(x) isthe inverse of lnx.Property 3 implies Property 5:dln(exp(x))dx=dxdx1exp(x)exp (x) = 1exp (x) = exp(x).For Property 6, letx= lna,y= lnb.Thenexp(x+y) = exp(lna+ lnb) = exp(ln(ab)) =ab= exp(x) exp(y).Note: a special case of Property 8 is(exp(1))y= exp(1·y)ey= exp(y).Chapter 6 Lecture NotesNine LecturesMAT186H1F Lec0107 BurbullaChapter 6: Applications of Integration6.1 Velocity and Net Change6.2 Regions Between Curves6.3 Volume by Slicing6.4 Volume by Shells6.5 Length of Curves6.6 Surface Area6.7 Physical Applications6.8 Logarithmic and Exponential Functions Revisited6.9 Exponential Models6.10 Hyperbolic Functions: An OverviewGeneral PowersFor anyb>0 we can definebx= (elnb)x=exlnb.Thendbxdx=dexlnbdx=exlnblnb=bxlnb,bxdx=exlnbdx=exlnblnb+C=bxlnb+C.Similarly, forx>0 we havex=elnx.This allows us to easilyhandle many computations with general powers and bases. Eg:dxxdx=dexlnxdx=exlnxxx+ lnx=xx(1 + lnx).Chapter 6 Lecture NotesNine LecturesMAT186H1F Lec0107 Burbulla
Chapter 6: Applications of Integration6.1 Velocity and Net Change6.2 Regions Between Curves6.3 Volume by Slicing6.4 Volume by Shells6.5 Length of Curves6.6 Surface Area6.7 Physical Applications6.8 Logarithmic and Exponential Functions Revisited6.9 Exponential Models6.10 Hyperbolic Functions: An OverviewNatural Growth EquationLetxbe the amount of some substance present at timet.Thefollowing differential equationdxdt=kx,k= 0has many important applications. It can be interpreted asdxdtthe rate ofchange=kis proportional toxthe amount present.
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