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` .**

## See eNotes Ad-Free

Start your **48-hour free trial** to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Already a member? Log in here.