Can we assume that if x>1 that ln(x) >x -1 (??)
therefore x > exp(x - 1) = exp(x) / e
x .1 means for all x there must exist p>0 such that x=1+p and substituting this for x we see that
1 + p > exp(1+p)/e = exp(p).e/e = exp(p)
Our assumption produces then 1+p > exp(p) and this is false if p=2,3,4,.......
So assumption is false, proving the result.