$2.55<a<2.85$, where $e^x$ intersects exactly once with $x^a$ whilst $x>0$. Find the value for $a$, at which this single intersection occurs. ($a$ must be a real number)
I would really appreciate it if somebody could post the solution to this problem, have been struggling to solve for a while.
I have tried solving $e^x = x^a$ by differentiation however i feel this is the wrong approach as i always end up with $x = a$ which sounds completely wrong.
You write yourself that after differentiating, you arrive at $x=a$. This does make sense, however: This is the value of $x$ that satisfy the equality, i.e., this is were the slopes of the two functions are equal.
Inserting into the original equation, we have $$e^x=x^a\rightarrow e^a=a^a,$$ which can only be satisfied if $a=e,$ which is indeed the answer.