Question

Solve the equation. Round to the nearest hundredth. Show work.


`6e^(2x) - 5e^(x) = 6`

Answer 1

The value of x = 0.41

Explanation:

∵
`6e^(2x)-5e^(x)=6`

Let
`e^(x)=y`

∴
`e^(2x)=y^(2)`

∴ 6y² - 5y = 6

∴ 6y² - 5y - 6 = 0 ⇒ factorize

∴ (3y + 2)(2y - 3) = 0

∴ 3y + 2 = 0 ⇒ 3y = -2 ⇒ y = -2/3

∴ 2y - 3 = 0 ⇒ 2y = 3 ⇒ y = 3/2

∵
`y=e^(x)`

∴
`e^(x)=(-2)/(3)` ⇒ refused

(
`e^(ax)` never gives -ve values)

∵
`e^(x)=3/2` ⇒ insert ln in both sides

∵
`lne^(ax)=axlne=ax` ⇒ ln(e) = 1

∴
`xlne=ln(3/2)`

∴ x = ln(3/2) = 0.41