If a>b, lna>lnb
ie, lna>lnb if a>b because derivative is pos.
Why make everything so complicated?
I give a 3 line answer to the OP and you turn a mole-hill into a mountain.
here. It is rude to hijack threads and troll those attempting to actually help.
e^x = 1+x+.... >1+x
Every step is trivial and would have been taken for granted in an introductory calculus course.
But I was asked to show: e^x>e^y -> x > y which I did by showing that
If a>b, lna>lnb which I considered trivial at this level of discussion (obviously derivative of ln>0 without a lot of blah-blah, or look at the graph). Actually, this step was hi-jacked by someone else with an exagerated discussion in abstract terms to prove this trivial statement.
What did I hi-jack? One of the other posts? you’re kidding.
And yet a statement like substitute into Napier's inequity and the answer falls right out goes unquestioned.
You are correct. Three sols were given. The first in the first post.
1) f(x) = x-ln(1+)x and f’(x)>0. Good answer.
2) Napier’s inequality. Answer
My only objection to this answer is it proves one unproved inequality with another unproved inequality. Couldn’t find any reference to it in a calculus book. I vaquely recall seeing it in statistics, so possibly inappropriate to subject. So the answer depends on happening to remember a formula from somewhere.
3) x>lnx, e^x>1+x, e^x=1+x+…>1+x. Answer
I missed 1) and 2) because I was thrown off by the endless pages of math which I had no inclination to follow and which led me to believe that the respondents were still fishing for an answer.
3) was posted after a considerable lull and then followed soon after by another huge chunk of math out of the clear blue sky in response to an unreferenced “hint.” I could only conclude without wasting a lot of time that the answer was still being sought, which puzzled me since an answer had just been given.
The suggestion that I hi-jacked an answer is outrageous. All three answers are clearly independent.
That really only leaves 3) in the spirit of the question which can be shortened to:
EDIT: OK, and the inverse (ln) of an increasing function is increasing. I like the essence of a solution without distractions. That's why I liked 1) before I realized it wasn't the essence of the solution. But at least it was clear.