A Difference of Logs

A difference of logs joins the same way, with the quotient rule: subtracting logs divides the insides. $\log_{2}\left(x+12\right)-\log_{2}\left(x\right) = 2$ becomes $\log_{2}\left(\frac{x+12}{x}\right) = 2$.

Undo the log by raising 2 to both sides: $\frac{x+12}{x} = 4$. A fraction equal to a number clears by cross multiplying, which gives $x+12 = 4x$. Then $12 = 3x$, so x = 4.

A single candidate still gets checked. With x = 4 the inputs are 16 and 4, both positive, and $\log_{2}\left(16\right)-\log_{2}\left(4\right)$ is 4 minus 2, which is 2. It holds.