Question

Solve exactly in terms of ln. Be sure to check solutions: e^(2x) - e^x - 12 = 0

Answer 1

okay. lets see.
substitute e^x = y
rewrite the expression in terms of y.

Answer 2

did you try?

Answer 3

y=e^(2x) - e^x - 12 ?

Answer 4

no,
lets say we substitute y wherever we see e^x.
what do you get?

Answer 5

y^2 - y - 12 = 0

Answer 6

substitution basically

Answer 7

yes, now do you see that it is a simple quadratic equation which can be factorized?

Answer 8

which would be (y-4)(y+3)

Answer 9

right. so what are the values of y?

Answer 10

y= 4 or -3

Answer 11

okay, now substitute back e^x for y and solve

Answer 12

just to confirm?

Answer 13

note that ln e^x = x

Answer 14

so 4^2 - 4 - 12 = 0

Answer 15

hmmm they both work

Answer 16

yeah. it is a quadratic equation. it has 2 solutions.

Answer 17

ln(4^2) - ln(4) - 12 ?

Answer 18

I apologize still trying to grasp cpncept

Answer 19

no.

Answer 20

y = 4 and y = -3
that means
e^x = 4 and e^x = -3
ln e^x = ln 4 and ln e^x = ln -3
x = ln 4 and x = ln -3

Answer 21

but we know that logrithms don't exist for negative numbers. so the answer is x = ln 4

Answer 22

and thats it?