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Natural logarithms
25 From qn 10.6, weknow that D (1/ n ) is an upper sum and D (
1/ n )is
a lower sum for
dx / x .
Deduce that
dx
x , for a
L
( a )
1.
26 Why must
L
( a )
0as a
1
, and
L
( a )
as a
? Why is
L
strictly increasing, continuous and L ( x ) 1/ x .
27 Why must there be a number e 1 such that L (e) 1? Check that
L
(2)
1, so that 2
e.
When a e, write A ( x ) E ( x ) exp( x ).
28 Provethat E ( x ) E ( x ). What is the Taylor series for E ( x )?
If wenow define A for 0 a 1, wecan usethefact that 1
1/ a
to apply thersults of qn 10 onwards. With this condition, in the
analogues of qns 10 and 21, A is then strictly decreasing. The analogue
of qn 13 holds for negative integers and negative rationals and the
analogues of qns 14 and 15 transpose the originals. The analogue of qn
16 holds, but for the analogues of qns 17 and 18 an interval [ c ,0]
must be used and the result extended from the negative reals to the
positive reals. The analogues of qns 20, 23 and 24 hold, but for qn 25
have D ( 1/ n ) as an upper sum and D (1/ n ) as a lower sum; so we get
L ( a ) as before when 0 a and L ( a ) as a 0 .
It is trivial to define A ( x ) 1 when a 1, and then L (1) 0.
Now L is defined on the domain R and is strictly increasing and
continuous.
29 By differentiating the function f L E , show that L ( E ( x )) x for
x R
.
30 Since
L
and E arebijctions, E (
L
( x ))
x for x R
. Deduce that
L ( x ) log
x .
Thefunction L is called the natural logarithmic function and is
usually denoted in school texts by ln. When log x is written without an
explicitly named base, in university texts, the logarithm to the base e is
meant. The function A is an exponential function, but when the
exponential function is referred to, it is E that is meant.
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