Question

Solve for e^2x-e^x=6

Answer 1

$e ^{2x} -e ^x=6$ ----(i)

Let, $e ^x =t$
Therefore, $e^{2x} = t^{2}$

Therefore, from (i),

t²-t=6 
t²-3t+2t-6=0
t(t-3)+2(t-3)=0

Therefore, (t-3)(t+2)=0
Therefore, t=3 or t = -2

But, 
$e^x =t$

Therefore, 

$e^x = 3$ or $e^x = -2$

Taking ln on both sides,
Therefore, 
$ln(e^x)=ln(3)$ or $ln(e^x) = ln(-2)$

But natural log of negative numbers does not exist since negative numbers are not in the domain of ln(x)
Therefore, 
$ln(e^x) = ln(-2)$ is discarded 
Therefore, x = ln(3) is the only solution.