How to solve exponential inequality with x I need to solve the following inequality. ln(x)−x>0. I oddly remember that it can only be done by using the graph... Is it true? I have the same problem with e^x(x−1)>−2. Thanks!
Dean Summers
Answered question
2022-07-20
How to solve exponential inequality with I need to solve the following inequality.
I oddly remember that it can only be done by using the graph... Is it true? I have the same problem with
Thanks!
Answer & Explanation
Danica Ray
Beginner2022-07-21Added 15 answers
Let . Note that is meaningful for . Then, so that for , the function is increasing and for , the function is decreasing. At , we have . Together, these last 2 sentences say is always less than or equal to . In other words, there is no real such that Similarly, let . Then so that decreases for negative and increases for positive . When , evaluates to . This means is always greater than or equal to . In particular, for all real ,
Hayley Bernard
Beginner2022-07-22Added 5 answers
These are both consequences of the standard inequality
How to best prove (1) depends on how you develop the theory of the exponential function. If you know that is its own derivative, then you can argue that
(because if and if ; note that we could have so that the integration interval is backwards). Anyway, once you have (1), take logs to get
and then replace with to get
which implies your first inequality. For the second, if then multiplying (1) by yields
which implies your second inequality in the case . On the other hand, if then replacing with in (1) yields
taking reciprocals (which reverses the inequality because both sides are positive in this case) yields