The standard log function form is $a \log[k(x-d)] + c$
Where
$a$ vertically stretches or compresses
$k$ horizontally stretches or compresses
$d$ translates left or right
$c$ translates up or down
If we have $2 \log(2x)$ does this have a vertical translation when rewriting to $\log 4 + \log x$?? And why? Standard form dictates this has no vertical or horizontal shifts?
When dealing with logarithms, vertical shifts, and horizontal stretches/compressions are considered to be one and the same.
$$\log(a\cdot x)=\log(x)+c$$
We don't say it doesn't have a vertical shift, it does, but it also has a horizontal stretch/compression that is exactly related to the vertical shifting.
To understand why this is so, return to the definition of the logarithm:
$$\log_{10}(x)=y$$
$$x=10^y$$
Now, try performing a horizontal stretch/compression by a multiple of $10$.
$$10x=10^y$$
$$x=\frac{10^y}{10^1}=10^{y-1}$$
Convert this back to logarithmic form:
$$y-1=\log_{10}(x)$$
$$y=\log_{10}(x)+1$$
In the beginning, we had $10x=10^y$, which in logarithmic form is $y=\log_{10}(10x)$:
$$y=\log_{10}(x)+1=\log_{10}(10x)$$
So that's the basic reason.
Vertical stretches/compressions are not synonymous:
$$a\log(x)=\log(x^a)$$
That is neither a shift nor a stretch, as you can obviously tell.