Properties of the natural logarithm and exponential functions

Transcription

1 Poeties of the natual logaithm an eonential functions Define fo ositive the function A() as the aea fom to une the hyeolay Since thee is no with, A() 0 By efinition the eivative of A() is given y the limit lim A + h A The numeato A( + h) A is the aea une y fom to h 0 h + h Bouning this aea etween the two ectangles shown gives the inequality h h A( + h) A that < A( + h) A < Hence, < < + h + h h Al Lehnen

2 Poeties of the natual logaithm an eonential functions Fom the squeeze theoem A In summay, A 0 Poety : Poety 2: A ( + ) A h A lim h 0 h Consie, fo an oth ositive, A( ) A A( ) A 0 Hence, 0 A A A A A must e a constant Since A, Poety 3: A( ) A( ) + A Hence, Consie, fo ositive, 2 A A( ) ( ) A A( + ) must e a constant A A( ) A A Poety 4: A( ) A Note: this means that fo A the aea is to the left athe than the ight of < <, < 0, this is inteete as a negative aea since Fom oeties 3 an 4, A A + A A A( ) Poety 5: A A A( ) Consie, fo ositive an a ational nume, A( ) A ( ) 0 Hence, must e a constant A( ) A A A A A Poety 6: A( ) A Now, A 2 > 2 Poety 7: lim A, so A 2 A 2 > as u Fom Poety 4, lim A lim A lim A( u) Al Lehnen 2

3 Poeties of the natual logaithm an eonential functions Poety 8: A lim 0 + The omain of A() is ( 0, ) an fom oeties 7 an 8 the ange of A() is (, ) Poety 9: Domain of A() is ( 0, ) an the ange of A() is (, ) name of A() is the natual logaithm, A The stana Thee is a value of to the ight of with an aea of une the hyeolay This value is given the symol, e This nume is igge than 2 ut smalle than 3 Poety 0: ( e ) > 0, () is an inceasing function so it must ass a hoizontal line test Thus, thee is an invese function which is esignate as e() Since Poety : Domain of e() is (, ) an the ange of e() is ( 0, ) Poety 2: e fo> 0; Poety 3: e( 0) ( 0) Poety 4: e( ) ( ) e e fo any eal nume Al Lehnen 3

4 Poeties of the natual logaithm an eonential functions Consie eal numes a an Fom Poety 3, ( ( )) ( ( )) e a+ e e a + e e e a e e a e Poety 5: e( a+ ) e( a) e( ) Consie eal numes a an Fom Poety 5, e( a) e e( a ) e( ( e( a) ) ( e( ) )) e e ( ) e e( a) Poety 6: e( a ) e ( a) ( ) Al Lehnen 4

5 Poeties of the natual logaithm an eonential functions Consie a eal numes a an a ational nume Fom Poety 6, ( ) ( ) e a e e a e e a e a Poety 7: e( a) e( a) Thus, the e function oeys the ules of eonents an hence is calle the eonential function Fom oety 7, ( ) e e e, since e() makes sense fo any eal nume we "eten" the notion of eonent to any eal nume with the following efinition which gives an altenate symolism fo the eonential function Poety 8: e e Using 8, the notion of an eonent can e etene fo any ositive ase,, to any eal e e e fo any ational nume an ( ) nume So fo any eal nume an ositive ase,, the eonential function ase is efine as follows Poety 9: e If e, y e ( ) ( ) e is a one to one function an has a unique invese given y ( y) ( y) ( ) ( e( )) ( ) so that the invese eonential function, ase, is given y the change of ase fomula as follows Poety 20: log ( ) f Using the funamental fomula that e e given y e ( ) / e Poety 2: e e e e, the eivative of e() is f f ( ) Al Lehnen 5

6 Poeties of the natual logaithm an eonential functions Fo any ositive ase,, ( ) e( ( ) ) e ( ( ) ) ( ) e( ( ) ) ( ) ( ) Poety 22: Fo any ositive ase,, l log Poety 23: log l Fo ositive the owe law function can now e efine fo any eal eonent ( ) e e e e Thus, ( ), an ( ) ( ) e e e e( ) e( ) ( ) e e Theefoe, the owe law ule of iffeentiation woks fo any eal eonent if the ase is ositive Poety 24: ( ) A moe useful efinition of e is foun in the following limit lim + Fom Poety 9, lim lim e + + Since the eonential function has a eivative it must e continuous so the limit can e taken insie the eonential Now, fom the efinition of + as the aea fom to + une the hyeolay < + < < + < Hence + + fom the squeeze theoem, lim + + e an lim e( ) Al Lehnen 6

7 Poeties of the natual logaithm an eonential functions Poety 25: lim e + A genealization of this esult fo any eal nume is the following limit lim + Again fom Poety 9, lim lim e + + Fom the efinition of + as the aea fom to + une the hyeolay allows the following ouns, < + < < + < Hence fom the + + squeeze theoem, lim + an taking the limit insie the eonential gives the esult that lim e( ) e + Poety 26: lim e + Al Lehnen 7