e^-x + 2e^x=3

Question

solve for x without estimating.

Answer 1

e^-x + 2e^x=3                  e^-x = 1/e^x

1/e^x + 2e^x = 3

1/e^x  + 2e^(2x)/e^x = 3

( 1 + 2e^(2x))/e^x = 3

 1 + 2e^(2x) = 3e^x

  2e^(2x) -  3e^x + 1 = 0

( 2e^x  - 1)(e^x - 1) = 0

2e^x - 1 = 0 ⇒ e^x = 1/2 ⇒ x = ln(1/2)

or

e^x - 1 = 0 ⇒ e^x = 1 ⇒ x = 0 because e^0 = 1