indications equation: a^x=x+2 has two real solutions.I need to find "a" values.

[Vali]

indications equation: a^x=x+2 has two real solutions.I need to find "a" values.

Hello!
a^x=x+2 has two real solutions.I need to find "a" values.
A) (1, infinity)
B) (0,1)
C) (1/e , e)
D) (1/(e^e), e^e)
E) (e^(1/e), infinity)
I tried to solve and I did it but I don't understand some things.
I let a picture below to see.
First, I need to know if there's other 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 decrease from infinity to -1, then increase from -1 to infinity.I'm really confused.Need some indications here.
Thank you!

 

[tkhunny]

Hello!
a^x=x+2 has two real solutions.I need to find "a" values.
A) (1, infinity)
B) (0,1)
C) (1/e , e)
D) (1/(e^e), e^e)
E) (e^(1/e), infinity)
I tried to solve and I did it but I don't understand some things.
I let a picture below to see.
First, I need to know if there's other 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 decrease from infinity to -1, then increase from -1 to infinity.I'm really confused.Need some indications here.
Thank you!

First, you can't "solve" it, so the claim that you did is a bit troubling.

Second, immediately rule out a = 0 and a = 1. Only one intersection.

Third, what do you think about a < 0? Does it make sense at all? What does it mean? Did the problem statement SAY a > 0?

Fourth, you seem to be on the right track. What does y = 0.5^x look like? Can you hit it twice with the line? What does y = 2^x look like?

 

[Vali]

First, you can't "solve" it, so the claim that you did is a bit troubling.

Second, immediately rule out a = 0 and a = 1. Only one intersection.

Third, what do you think about a < 0? Does it make sense at all? What does it mean? Did the problem statement SAY a > 0?

Fourth, you seem to be on the right track. What does y = 0.5^x look like? Can you hit it twice with the line? What does y = 2^x look like?

Yes, so sorry but the statements are in romanian and I need to translate it.I need to find positive values of "a" such that a^x=x+2
Sorry for the term "solve" I'm not familiar with english terms.
In this kind of exercises I usually treat a>1; a<1; 0<a<1 ; a=1; a=0

 

[Steven G]

Yes, so sorry but the statements are in romanian and I need to translate it.I need to find positive values of "a" such that a^x=x+2
Sorry for the term "solve" I'm not familiar with english terms.
In this kind of exercises I usually treat a>1; a<1; 0<a<1 ; a=1; a=0

No need to look at a<1, just consider 0<a<1.
Your graphs are for a>1 and 0<a<1. What about the other cases YOU listed? Also please graph the line y=x+2 on all graphs and see if y=a^x ad y=x+2 intersect.

 

[Vali]

The graphs intersects in 2 points just when a>1
for 0<a<1 y1=a^x and y2=x+2 intersect in one point ( same for a=1)
So a>1