Trying to prove that for x real, ln(1+x)<=x

[penguin007]

Homework Statement

Hi everyone,
I'm trying to proove that for x real, ln(1+x)<=x

Homework Equations

The Attempt at a Solution

I know that I can solve this problem by introducing f:=x->ln(1+x)-x and studying f, but I'd like to use x->ln(1+x)'s concavity so I tried to find lambda, x and y in :
ln(1+(lambda)*x+(1-lambda)*y)>=(lambda)*ln(1+x)+(1-lambda)*ln(1+y)...


Any help is welcome...

 

[l'Hôpital]

You are trying too hard.

Do you know the taylor series expansion for e^x? That is all you need.

 

[penguin007]

I find your solution quite interesting too (I had not thought about it) but I'd like to find a solution using the concavity of this function x:=->ln(1+x) (though I admit it may be too hard for that question).