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Prove that log(e)(1+x)ltxforxgt0...

Prove that `log_(e)(1+x)ltxforxgt0`

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Let `f(x)=log_(e)(1+x)-x`
`therefore f(x)=(1)/(1+x)-1=-(x)/(1+x)`
For `xgt0,f(x)lt0`
So ,f(x) is strictly decreasing function for `xgt0`
`therefore f(x)ltf(0)forxgt0`
or `log_(e)(1+x)-xlt0`
`therefore log_(e)(1+x)ltxfor xgt0`.
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