How do I solve the equation e^(2x)-2=e^x?

[Jaco Viljoen]

Homework Statement

h(x)=e^(2x)-2
a)Write down the Dh
b)Solve the equation e^(2x)-2=e^x

Homework Equations

The Attempt at a Solution

a)x∈ℝ[/B]
e^(2x)-2=e^x
((e^x)^2)-e^x-2=0I don't know what to do here:

 

[Mark44]

Homework Statement

h(x)=e^(2x)-2
a)Write down the Dh
b)Solve the equation e^(2x)-2=e^x

Homework Equations

The Attempt at a Solution

a)x∈ℝ[/B]
e^(2x)-2=e^x
((e^x)^2)-2=e^x

I don't know what to do here:

Move the e^x term to the left side. Your equation is quadratic in form, and can be factored. Keep in mind that e^x > 0 for all x.

 

[Jaco Viljoen]

Hi Mark,
Thank you,
I initially did this but how do I factor e^(x2)
I also considered adding a ln to both sides to get rid of the e on both sides but the -2 is the problem.

 

[Ray Vickson]

Hi Mark,
Thank you,
I initially did this but how do I factor e^(x2)
I also considered adding a ln to both sides to get rid of the e on both sides but the -2 is the problem.

You need to go back and review the laws of exponents. How does $e^{2x}$ relate to $e^x$? Surely you must have had such material already, or else you would not have been asked to do this problem.

 

[Mark44]

Hi Mark,
Thank you,
I initially did this but how do I factor e^(x2)

What you wrote in post #1 should be something of a hint.

e^(2x)-2=e^x
((e^x)^2)-e^x-2=0

I also considered adding a ln to both sides to get rid of the e on both sides but the -2 is the problem.

If you have terms added together, taking the log is not any help.

 

[Jaco Viljoen]

((e^x)^2)-2=e^x
(k^2)-k-2=0 k=e^x
(k+1)(k-2)=0
k=-1 or k=+2

e^x=2
lne^x=ln2
x=ln2

 

[Mark44]

((e^x)^2)-2=e^x
(k^2)-k-2=0 k=e^x
(k+1)(k-2)=0
k=-1 or k=+2

e^x=2
lne^x=ln2
x=ln2

This is correct, but you should say why your are discarding the other value of k.

 

[NicolasPan]

1.a) Dh=ℝ
b)The solution can go through this path: (e^x)^2-e^x-2=0
If e^x=y i) then the equation is transformed like that: y^2-y-2=0
then the solutions are y=(1±3)/2 and from i) we get e^x=(1±3)/2 BUT since the function e^x can only give positive results for x∈ℝ the only acceptable solution would be the one with the +. So e^x=2 ⇔ x=ln2, there you go friend I hope this helps :)

 

[Mark44]

1.a) Dh=ℝ
b)The solution can go through this path: (e^x)^2-e^x-2=0
If e^x=y i) then the equation is transformed like that: y^2-y-2=0
then the solutions are y=(1±3)/2 and from i) we get e^x=(1±3)/2 BUT since the function e^x can only give positive results for x∈ℝ the only acceptable solution would be the one with the +. So e^x=2 ⇔ x=ln2, there you go friend I hope this helps :)

The OP has already gotten all of this.