# Math Help - e and ln

1. ## e and ln

I have half done this question and icant seem to finish it im completely suck.. so any help is appreciated

Find the exact values of x for which...

(e^x)/(((e^x)-2)^2) = 1

2. Originally Posted by pop_91
I have half done this question and icant seem to finish it im completely suck.. so any help is appreciated

Find the exact values of x for which...

(e^x)/(((e^x)-2)^2) = 1
It's refreshing to see a unambiguously written equation

$\frac{e^x}{(e^x -2)^2} = 1$

$e^x = (e^x-2)^2$

Expand the right hand size as per the binomial theorem/normal quadratic

$e^x = (e^x)^2 - 4e^x + 4$

$(e^x)^2 - 5e^x +4 = 0$

This is a quadratic in $e^x$ and so the quadratic equation can be used - however this one factorises

$(e^x-4)(e^x-1) = 0$

Spoiler:
$e^x = 4 \: \rightarrow \: x = 2ln2$

$e^x = 1 \: \rightarrow \: x = 0$