calc help!?

Question

if f(x)= e^2x - 5e^x + 6=0 has 2 solutions,
what is the smaller solution?
what is the larger solution?

Answer 1

Let y = e^x
y² - 5y + 6 = 0
(y - 3)(y - 2) = 0
y = 3 , y = 2
e^x = 3 , e^x =2
x = ln 3 , x = ln 2
x= ln 3 is the larger solution.

Answer 2

Do a substitution: u=e^x.

Then your equation becomes u^2 - 5u + 6 = 0.

Solve for u; you get -1 and 6.

So -1 = e^x and 6 = e^x, replacing the values in the substitution.

e^x cannot equal -1, so that can't be a solution.

For the other one, take the ln of both sides and you get x = ln(6).

Answer 3

factor it ...e^x=3, or e^x=2 etc
x=ln3 or x=ln2
[(e^x-3)(e^x-2)=0 is e^2x-5e^x+6=0]

Answer 4

Como's solution is correct

Answer 5

use the discriminant