Comparing function growth of exponentials with a different base

Question

I'm having a hard time determining the order of growth rates of these functions:

$2^x$

$3^\frac{x}{2}$

$x*2^x$

Answer 1

Obviously $x* 2^{x}$ grows faster then $2^x$ (because the question was about the growth of the function, I am not treating the case when $x<1$).

And I will affirm that $3^{\frac{x}{2}}$ grows slower than $2^x$ and that is due to the fact that: $3^{\frac{x}{2}} = \left( 3^{\frac{1}{2}}\right) ^x = \left(\sqrt{3}\right) ^x < 2^x$.

Hence, $x* 2^{x}$ grows faster then $2^x$, which grows faster then $3^{\frac{x}{2}}$.