Answer to Question #177916 in Real Analysis for Nikhil Singh

Question

Prove that

x< log(1/1-x)< x/1-x ; 0<x<1

Answer 1

$\bigstar$ To show

x< log(1/1-x)< x/1-x ; 0<x<1

We can prove this by using

$\bigstar$ Mean Value Theorem

for all x>0

$\bull$ using the mean value theorem

$\boxed{x< log({1\over1-x})}$

Here x can't equal to 1

Let

$\boxed{f(x)=x - log({1\over1-x})}$ ..............1

Since

f(0) = 0

$\boxed{f'(x) = 1+{1\over(1-x})}$ .............2

for all x>0

 f(x)<0

Now

$\boxed{log({1\over1-x})<{ x\over1-x }}$ ............3

let

$\boxed{f(x) = log({1\over1-x}) - {x\over1-x }}$ ............4

$\bigstar$ according to the mean value theorem, an x₀

between a and x

with

$\boxed{f'(x_o ) = { f(x)-f(a)\over x-a}}$ ...........5

$\bull$ Using the property

, we can prove the inequality

By using (1) ,(2) ,(3), (4) ,(5)

We got

$\bigstar$

$\boxed{ x< log({1\over1-x})< {x\over1-x }; 0<x<1}$