Exponential algebra problem We need to solve for x:
$$54\cdot 2^{2x}=72^x\cdot\sqrt{0.5}$$
My proposed solution is below.

I think this is a more systemactic way:
$$54\cdot 2^{2x}=72^x\cdot\sqrt{0.5}$$
Apply logarithm on both sides of the equation. For now the base of the logaritm does no really matter
$$\log{(54\cdot 2^{2x})}=\log{(72^x\cdot\sqrt{0.5})}$$
and simplify by applying  the laws of logarithm for  products and powers 
$$\log{(54)} + 2x \log{(2)}=x\log{(72)} + \log{(\sqrt{0.5})}$$
to get a  linear equation in x. Now solve this equation :
$$x=\frac{\log{(\sqrt{0.5})}-\log{(54)} }{2\log{(2)} - \log{(72)}}$$
Now you can try to simplify this expression for $x$.
