Geometric mean is the (arithmetic) mean of magnitudes

For a set $$\begin{gathered} A= \\ {x}_1,x_2,...,x_n \end{gathered}$$ of positive numbers, the arithmetic mean is defined as $\frac{1}{n}\sum x_i$ whereas the geometric mean is defined as $\sqrt[n]{\prod x_i}$, resulting in:

$A$	Arithmetic Mean	Geometric Mean
$\{4,9\}$	$6.5$	$6$
$\{1,2,4,8,16\}$	$6.2$	$4$

The Geometric Mean becomes obvious if we write all numbers as magnitudes:

$A$	Arithmetic Mean	Geometric Mean
$\{2^2,2^{3.1699}\}$	$2^{2.7004}$	$2^{2.58496}$
$\{2^0,2^1,2^2,2^3,2^4\}$	$2^{2.632268}$	$2^2$

Note that $2.58496$ is just the Arithmetic Mean of $\{2,3.1699\}$, and for the second line $2$ is the mean of $\{0,1,2,3,4\}$. I have written them base $2$ but any magnitude would suffice.