/
The
ngmber e
-The
irrational
number
e
(x
2.72)
is
the
base for natural
logs
-The
number
e
is
defined
in
calculus
as, lim
(1
*
l/n)n
:
e
*Maior
proue4ie$,of
loss!
The following
hold
for
any base
a. I will
state them
using
the
In
(base
e)
1) ln(uv): In
u
+
lnv 2)
ln(u/v):
lnu
-
Inv
3)
ln(u')
=
n
lnu.
Give examples
-elgggg-At-Usggf
Any
base
log may
be wriuen
in
terms
of any other
base. In
the fornnrla
below,
a base
log
is
written in
terms
of base
e
(
or the ln).
*LOgrA:
hA
whereB
>0
andA>0
Give
example:
lnB
-Log
(base
10)
and
ln
(base
e)
are buttons
on
your
TI-83...take
a
look.
-The
tog
of
a
positive
number
A to
that
base A is
always
equal
to 1
dog
oA:
l),
so,
In
e
=
l.
Exafirp]e:
Solve for
t
using natural logs:
40
-
100
e-'03'
Growth
lYpdpl$
-annuat
erowth
(oYer
I
year):
Let
P"be
the original
amount that
is
growing
atf/o
per
year.
Then
Pogrows
to
Po*rPo
at
the
end of 1
year.
This
can be
ruritten
as
P"(l+r),
where r7o
is
erpressed
as a
decimal.
After
2
years,
we
get
Po(1+r)(1+r1
=
Po(l+r)2
,
Continuing
in this
way,
after t
years,
we
will have P"(t+r)'.
@
uses t.he model Poe'r.
It
is important
to note
that
an annual
growth
rate
of
f/o,@.
sgl
yield
&e same resrlt
as
a
cootinuous
growth
rate ofrYo-(wi[
give
details
later
on
compound
interest)
-----------]
Example:
A
populafiorl
P, is
growing
at a6Yocontinuous
rate,
Find
the
equivalent
annual
rate and write
an
equation for
P using both
types
of
growth.
