Find the exact solution for e⁽²ˣ⁾-2e⁽ˣ⁾-63=0

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23-06-2024
Mathematics

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Find the exact solution for e⁽²ˣ⁾-2e⁽ˣ⁾-63=0

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olumideolawoyin olumideolawoyin · Answer 1 · 2024-06-23T09:33:18-04:00

Answer:

x = 2.1972

Step-by-step explanation:

[tex] {e}^{2x} - 2 {e}^{x} - 63 = 0[/tex]

Let

[tex] {e}^{x} \: be \: y[/tex]

[tex] {e}^{x(2)} - 2 {e}^{x} - 63 = 0[/tex]

y² - 2y - 63 = 0

Using Factorization method

y² - 9y + 7y - 63 = 0

y(y - 9) + 7(y - 9) = 0

(y - 9)(y + 7) = 0

y - 9 = 0 , y + 7 = 0

y = 9 , y = - 7

Therefore,

[tex] {e}^{x} = 9 \: and \: {e}^{x} = - 7[/tex]

Take In of both sides

[tex]In( {e}^{x} ) = In9[/tex]

x Ine = In 9 Applying logarithm law of same base.

x = In 9

and

[tex]In {e}^{x} \: = In( - 7)[/tex]

x Ine = In(-7) Applying logarithm law of same base

x = In (-7) is undefined.

Hence,

x = In9

x = 2.1972