Find the number of elements $a^{x}=x$ depending on $ a> 0 $ My tryLet $$f(x)=a^{x}-x=e^{x\cdot ln a}-x$$So $$f'(x)=a^{x-1}\cdot x-1$$Then I should examine the $ f $ monotonicity.But when I can do it I have:$$f'(x)>0$$$$a^{x-1}\cdot x>1$$However I don't know what can I do in this moment. I tried:for $x>0$: $(x-1)lna>-lnx$ but I still do not know how to get to zero places $ f '(x) $.Can you get some tips how do it?

You have differentiated $a^{x}$ wrongly. It should be $a^{x}\ln(a)$. But I suggest you sketch the graphs of $y=x$ and $y=a^{x}$ on the same axes and consider the number of intersections, and the conditions on $a$ for there to be any intersections at all.
