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```
If y=log e x then for x>1, show that we must have (a) x-1>y (b) x 2 -1 >y (c) (x-1) / x

```
11 years ago

```							 hi,
if two functions are equal at a point , & after that point if one has always a high derivative than other , it means that it increases  more with the same change in x than other,it clearly emplies that it would always be high than other
(a) x-1>y
at x =1 ,
l.h.s.=r.h.s . = 0
d/dx(l.h.s) = d/dx(x-1) =1
d/dx(r.h.s) = d/dx (log e x) = 1/x
if x >1,
1/x <1
d/dx(l.h.s) > d/dx(r.h.s)
emplies,
x-1>y

(b) x2-1 >y  ,                at x=1
l.h.s = rh.s = 0
d/dx(l.h.s) =d/dx (x2-1)
= 2x
d/dx(r.h.s) =d/dx (log e x ) =  1/x
we know,
for x>1 ,
2x >1/x
hence,
x2-1 >y

(c)  (x-1) / x <y
at x= 1,

l.h.s = rh.s = 0,

d/dx(l.h.s) =d/dx ( (x-1) / x )
= 1/x2
d/dx(l.h.s) =d/dx (log e x )
= 1/x
for x>1 ,
1/x2 <1/x, hence
(x-1) / x <y
```
11 years ago
```							hi tushar,

x = e^y , now for 1st case substitute e^y in place of x and u will get (e^y - 1) > y, here u can see that the function on the left hand side is  a exponential function and for x>1, e^y > 0 ,also for any value of y , the left side fxn will increase with a slope of e^y while right side fxn will increase only with a slope of 1 so, for all value of y ,(e^y - 1)>y........

similarly for 2nd case the eqn is (e^2y - 1)> y ,we can say that it is always valid for x>1......

now in 3rd case , ur eqn wud be (1 - 1/e^y) here u can see its slope is 1/e^y , so its slope is always less than 1 for all x>1....so the given condition (x-1) / x

```
11 years ago
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