The number of real roots of the equation $5+|2^x-1|=2^x(2^x-2)$ I'm trying to find the number of real roots of the equation $5+|2^x-1|=2^x(2^x-2)$.
Let $2^x=a$
$$|a-1|=a^2-2a-5$$
Then there are two cases 
$$a-1=a^2-2a-5$$
And $$a-1=-a^2+2a+5$$
Solving both equations 
$$a=1,-4,-2,3$$
Now -4 and -2 can be neglected so there are two values 
1 and 3.
Then $$2^x=1$$
$$x=0$$
And $$2^x=3$$
$$x=\log_2 3$$
But the answer doesn’t seem to consider the $\log_2 3$ as a viable root, and the answer is 1. Why is that the case?

Your way is right indeed for $2^x \ge1$
$$5+|2^x-1|=2^x(2^x-2)\implies 5+2^x-1=2^{2x}-2\cdot 2^{x} \iff2^{2x}-3 \cdot 2^x-4=0$$
for $0<2^x <1$
$$5+|2^x-1|=2^x(2^x-2)\implies 5-2^x+1=2^{2x}-2\cdot 2^{x} \iff2^{2x}- 2^x-6=0$$
then let $2^x=t>0$ and solve keeping only the solutions which agree with the assumptions.
Note that $2^x=3$ is not valid since it was obtained under the assumption that  $0<2^x <1$.
