Question

Question #60774

Answer 1

`x = (5+5e^4)/(e^4-1)~~5.187`

Explanation:

We will use the following properties of logarithms :

• `ln(a)-ln(b) = log(a/b)` for `a, b > 0`
• `e^ln(a) = a`

with these, we have

`ln(x+5)-ln(x-5) = 4`

(note that as the `ln` function only takes positive values, we must restrict `x` to be greater than `5`, or else we would have `x-5 < 0`)

`=> ln((x+5)/(x-5)) = 4`

`=> e^(ln((x+5)/(x-5))) = e^4`

`=> (x+5)/(x-5) = e^4`

`=> x + 5 = e^4x - 5e^4`

`=> e^4x - x = 5 + 5e^4`

`=> (e^4-1)x = 5+5e^4`

`:. x = (5+5e^4)/(e^4-1)~~5.187`