If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operat…

Let’s try this again …
(This is offered as a separate answer from my first, because it proposes something different.)
First, a bit of a digression: There’s a slight difference in “feel” with notation for products and fractions. The expression “$x \cdot y$” asks directly “What is the result of multiplying $x$ and $y$?”, which amounts to a straightforward computation. On the other hand $z/y$ –that is, the “inverse with respect to multiplication by $y$”– asks indirectly “What value, multiplied by $y$, yields result $z$?”
Of course, the fraction “$z/y$” admits a handy interpretation as a straightforward computation: “What is the result of dividing $z$ by $y$?” … although, when you really look at it, the computation has subtle alternative flavors: “Dividing $z$ into quantity-$y$ pieces yields a piece of what resulting size?” and “Dividing $z$ into size-$y$ pieces yields what resulting quantity?” This ambiguity is the result of the convenient commutativity of products: Since “$x \cdot y$” and “$y \cdot x$” amount to the same thing, it doesn’t matter which number corresponds to “size” and which to “quantity”. Despite the ambiguity, we somehow survive.
Now, with powers and roots and logarithms, we have same difference in “feel” … but since the “direct” computation (“this, to that power”) lacks commutativity, the flavors of the “indirect” inverse operations aren’t so subtle; moreover –and more importantly– those operations lack an intuitive(!) computational interpretation akin to “dividing” for fractions. (We often represent fractions with pizza slices; what’s the pizza-slice picture for a fifth-root? Of a log-base-7?)
The point of all this is that it may be helpful to devise a notation that amplifies the direct-vs-indirect dichotomy, to try and make clear when the numbers in the notation provide pieces of a computational result, and when they express a puzzle in terms of the a result and one of the computational pieces.
For example, I’ll keep the power notation from my previous answer:
$$x \stackrel{y}{\wedge}$$
This represents a direct computation: “$x$ raised to power $y$”. The left-to-right nature of the symbol is important, for the proposed inverse (with respect to $y$) would appear as
The interpretation here –again reading left-to-right– is that “(an implicit something) raising to power $y$ yields result $z$”. This is the $y$-th root of $z$.
For exponentiation and logarithms, we could start with …
$$y \underset{x}{\wedge}$$
… for the direct computation “$y$, raising base $x$”, and then …
$$\underset{x}{\wedge}\; z$$
… for the indirect puzzle: “(and implicit something) raising base $x$ yields result $z$”. This is the logarithm-base-$x$ of $z$.
That is, $\stackrel{y}{\wedge}$ always represents “raising to power $y$”, and $\underset{x}{\wedge}$ always represents “raising base $x$”. When these symbols are placed to the right of an argument, the argument is a part of a direct computation; when the symbols are place on the left of an argument, that argument is the result of a direct computation.
Although the notation succeeds in distinguishing direct and indirect concepts, I’m not really satisfied with it. The fact that $x^y$ is expressed in two different ways –$x\stackrel{y}{\wedge}$ and $y\underset{x}{\wedge}$– is strange; and the canceling inverses doesn’t seem as clean as it could be.
We could agree that down-arrows are inverses of up-arrows and leave things on the right:
x \stackrel{y}{\wedge} &\hspace{0.25in}\leftrightarrow\hspace{0.25in}& \text{$x$ raised to power $y$} \\
z \stackrel{y}{\vee} &\hspace{0.25in}\leftrightarrow\hspace{0.25in}& \text{$z$ resulting from raising to power $y$} \\
y \;\underset{x}{\wedge} &\hspace{0.25in}\leftrightarrow\hspace{0.25in}& \text{$y$ raising base $x$} \\
z \;\underset{x}{\vee} &\hspace{0.25in}\leftrightarrow\hspace{0.25in}& \text{$z$ resulting from raising base $x$}
This way, inverses cancel and commute (disclaimers apply) more cleanly, as in my first answer, though we still have distinct ways of expressing $x^y$. It’s a little weird to use down-arrows in notation that gets read in terms of “raising”, but perhaps all that’s needed there is a better symbol.