Find x in 3e^-x - e^x = -2 I really really need an answer ASAP.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to substitute `1/e^x`  for `e^(-x)`  in the given equation such that:

`3*(1/e^x) - e^x = -2`

You need to move all terms to one side such that:

`3*(1/e^x) - e^x + 2 = 0`

You need to bring all terms to a common denominator such that:

`3 - e^(2x) + 2e^x = 0`

You need to rearrange the termsin descending powers of x such that:

`- e^(2x) + 2e^x + 3 = 0`

You need to multiply by -1 such that:

`e^(2x)- 2e^x- 3 = 0`

You need to substitute t for `e^x`  such that:

`t^2 - 2t - 3 = 0`

You should use factorization such that:

`t^2 - 2t - 2 - 1 = 0`

You need to group the term such that:

`(t^2-1) + (-2t-2) = 0`

Converting the difference of squares `t^2-1`  into a product yields:

`(t-1)(t+1) - 2(t+1) = 0`

Factoring out`t+1`  yields:

`(t+1)(t-1-2) = 0`

You need to solve for t the following equations such that:

`{(t+1=0),(t-3=0):} => {(t=-1),(t=3):}`

You need to solve for x the equations such that:

`e^x = -1`  invalid => there is no`x in R`  for `e^x = -1`

`e^x = 3 => ln e^x = ln 3 => x ln e = ln 3 => x = ln 3 `

Hence, evaluating the solution to the given exponential equation yields `x = ln 3` .

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