Consumer Price Index - Index Numbers, Business Mathematics and Statistics Video Lecture | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year

previously we have seen log log X as a
function natural log okay log X as a
function and we saw the linear
approximation of log X at X naught equal
to one near one and why did I do that
duh I was looking at log X near one
because to start off with I knew
something about the normal one okay I
knew that log 1 equals to zero and hence
I could move forward okay now I can
always change my base point my base
point here was X naught which equals to
one now I want to probably check it at
another base point now what would that
base point be what would that base point
be where I could check you know natural
log function itself checking log X
checking natural log X at X not equal to
one gave me the following approximation
gave us the forward following
approximation which was log x equals two
X minus one around zero okay log x
equals two X minus one around zero now
taking this X minus one as some number Y
okay taking this X minus one as some
variable a number called Y ok so that
means X becomes Y plus one x becomes y
plus one earlier my base point was one
earlier my base point was one now what I
can do is I can take a new base point Y
to be zero and check accordingly and
check accordingly and it will be almost
same like checking at the base point one
which will give me the previous base
point x equals to 1 because why I am put
taking as a base point which equals to
zero ok let me just tell you it
this way so in case I take Y y equals 2x
minus 1 and y -1 will become X so that
means that you know natural log Y
natural log Y X will become sorry y plus
1 so natural log y plus 1 then equal to
Y can you see that it's the same thing
as writing let me just see in the color
Ln x equals 2x minus 1 what I've done is
in place of X I have written y plus 1
and in place of X minus 1 I've written Y
so now I can change my base point and I
can change my base point to 0 resulting
in the same thing so I just changed the
function so what I've done is we've just
shifted things we've just shifted things
by one we've just shifted things by one
that is it that's all I'm doing in the
domain I'm shifting things by 1 and
hence I'm getting a new functional value
approximation ok so basically what we
are trying to say is that log now I can
write it in terms of X now right I can
write it in terms of X of course and
what I get is log Ln or the natural log
of 1 plus X so that's my new function
now that's my new function now so what I
require is F F - and since I am looking
at the point 0 now I want F at 0 and s -
and 0 as well so now function is instead
of log X it is log 1 plus
it's okay so f dash will be 1 upon 1
plus X okay if 0 will be log of 1 which
is 0 and f dash X will be 1 upon f dash
X will be 1 upon 1 which is 1 so my
approximation will be Ln 1 plus X is
approximately not exactly equal it
should be approximately equal to F 0
plus F dash 0 X correct so that is equal
to 0 plus 1 into X which is X so
conceptually here I just told you how
you can shift things and mathematically
you just see how exactly you get the
same thing okay