Let’s make this a little bit easier by defining y=2^x. Then your problem reads

(y-1/y) / (y+1/y)=1/3

Normally, multiplying the top and bottom of a fraction by a variable is dangerous if the variable can be 0. But since 2^x can never be 0, we can do this with impunity. So, we multiply top and bottom by y to get.

(y^2 -1) / (y^2 +1)=1/3

Now it’s just a game of algebra. We multiply both sides by 3(y^2 +1) to remove the fractions and then solve:

3(y^2 -1)=(y^2 +1)

3y^2 -3=y^2 +1

2y^2 =4

y^2 =2

y=sqrt(2)=2^(1/2)

In that last line, I used the fact that taking the square root is the same as raising both sides to the 1/2 power. Now we use our definition for y to get

2^x =2^(1/2)

By inspection (or logarithms), x=1/2