The power of x

Answer: $x=3$

Using numbers

$x-1$ and $x+1$ are $2$ apart

Powers of $2$ (two columns so that numbers above each other are $2$ powers apart):

 $2$        $4$

 $8$       $16$    $4$ and $16$ are $12$ apart

$32$      $64$    numbers are getting too far apart

$16-4=12$

$2^4-2^2=12$

$x=3$

Using index laws

$2^{x+1}=2^x\times2^1=2^x\times2$ and $2^{x-1}=2^x\times2^{-1}=2^x\times\frac12$.

Substitute then factorise: $$\begin{array}{rl}{2}^{x+1}-2^{x-1}=& 12\\ \Rightarrow 2^x\times 2-2^x\times \frac{1}{2}=& 12\\ \Rightarrow 2^x(2-\frac{1}{2})=& 12\\ \Rightarrow 2^x\times \frac{3}{2}=& 12\\ \Rightarrow 2^x\times 3=& 12\times 2\\ \Rightarrow 2^x=& 24÷3\\ \Rightarrow 2^x=& 8\end{array}$$ So $x=3$, since $2^3=8$.