How to solve an exponential equation

Question

$8^X + 7 (2^{X+1}) = 7 (4^X) + 8$

$2^{3X} + 7 (2^{X+1}) = 7 (2^{2X}) + 2^3$

$3X = 3$

$X =1$

OR

$X+1 = 2X$

$X=1$

BUT answer $X = 0$, or $1$ or $2$ ????

Answer 1

I guess you compare $2^{3X}$ with $2^3$ to get $3X=3$ and compare $2^{X+1}$ with $2^{2X}$ to get $X+1=2X$, but you cannot do this in general.

Let $t=2^X$. Then, noting that $$2^{3X}=(2^X)^3=t^3,\ 7\cdot 2^{X+1}=7\cdot 2\cdot 2^X=14t,\ 2^{2X}=(2^X)^2=t^2,$$we have $$t^3-7t^2+14t-8=0\Rightarrow (t-1)(t-2)(t-4)=0\Rightarrow X=0,1,2.$$