For which $a$ does the equation $a^x=x+2$ have two solutions? I need to find values of $a$ for the following equation to have two real solutions.
$$a^x=x+2$$


*

*$(1,\infty)$

*$(0,1)$

*$1/e,e$

*$(1/(e^e), e^e)$

*$(e^{1/e}, \infty)$
This is how I solved this exercise, but I don't understand some things.

I would like to know if there's another way to solve this kind of exercise. I would be happy if I would get some ideas.
Also, from my solution, I don't understand why from that table results just one solution and from the graphic results two solutions. Usually, to see the number of solutions I use this kind of table.
For $a>1$, $f$ decreases from infinity to -1, then increase from -1 to infinity. I'm really confused. Need some suggestions here.
Thank you! 

try and draw to graphs, the first:$$f(x)=a^x$$
and the second:
$$g(x)=x+2$$
I recommend using www.desmos.com
now check where the 2 graphs meet
this is a case of Transcendental equation, I don't think it has an analytic solution, usually you solve them numerically  or graphically, or even using taylor series for approximation
