Mathematics: Solve the equation. round to the nearest hundredth. show work.

Solve the equation. round to the nearest hundredth.

devinb18

09.01.2022

The value of x = 0.41

Step-by-step 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

∵ $y=e^{x}$

∴ $e^{x}=\frac{-2}{3}$ ⇒ 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

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