### Preamble

The question asked is "Has anyone ever considered alternative notation?" I think that it almost certain that many people have, but it is equally certain that no such notation has caught on. Most of the other answers here discuss proposed notations, none of which seem to have any traction (or even *use*) in the wider mathematical community.

As such, I am presenting this answer as a frame challenge, which is meant to address what I perceive as the underlying problem, as well as some of the misconceptions which have prompted this question.

### Misconceptions in the Question

In the question, it is asserted that if $x^y = z$, then "we know that"
$$ x = \sqrt[y]{z} \qquad\text{and}\qquad y = \log_x(z). $$
This is incorrect.

If we assume that $x$ is a real variable, that $n$ is a given natural number, and that $a$ is a real number, then the equation
$$ x^n = a \tag{1}$$
has $n$ complex solutions. If $n$ is odd, one of those solutions will be real; if $n$ is even and $a > 0$, then two of those solutions will be real. The notation $\sqrt[n]{a}$, depending on context, either denotes the real solution to the (1) (if $n$ is odd), the nonnegative real solution to (1) (if $n$ is even and $a > 0$).

It is not obvious what $\sqrt[n]{a}$ should mean if $n$ is not a positive integer, though there are reasonable ways of defining this notation in terms of (1). For example, if $n$ is a natural number, we could define
$$ \sqrt[-n]{a} = a^{-n} = \frac{1}{a^n},$$
but we don't typically do that, and instead rely on exponential notation alone.

More generally, if $x$ is a complex variable, and $a$ and $n$ are complex constants, then $\sqrt[n]{a}$ typically denotes the *principal $n$-th root of $a$*, which can be defined in couple of slightly different ways, but generally means something like $r \mathrm{e}^{i/n}$. However, in this setting, there is enough ambiguity that one would usually choose to use more explicit notation in terms of the complex logarithm / exponential.

If we assume that $y$ is a real variable and that both $a$ and $b$ are positive real constants, then the equation
$$ b^y = a \tag{2}$$
has a unique solution:
$$y = \log_b(a) = \frac{\log(a)}{\log(b)}, $$
where $\log$ denotes the natural logarithm (or the common logarithm, or any logarithm with a fixed base—it doesn't really matter). On the other hand, as soon as we allow either $a$ or $b$ to be something other than a positive real number (say, a negative real number, or a complex number), things immediately become much more complicated, and (2) has many solutions (or none at all, depending on context).

In either case, we can make sense of the assertions stated in the original question *if* we first restrict the sets of numbers we are willing to consider. The suite of conclusions stated hold if we assume that $x$ and $y$ are positive real numbers and that $n$ is a natural number.

### History

It is worth noting that notations adopted here come from a few very different historical antecedents. It is only in relative recent mathematical history that the link between exponential, logarithmic, and radical functions was well understood and articulated. The notation reflects this.

While I am not an expert in this area, my understanding is that something like the following is true:

Radical notation stems from classical Greek thinking (this is not to say that the Greeks used this notation; only that it reflects their way of thinking about problems). To the Greek way of thinking, a number represented a physical quantity—a length, or an area, or a volume. A number is inherently represented by a length, multiplication of two lengths gives an area, and so on. In this mode of thinking, it is very reasonable to ask "If the area of a square is $a$, what is the length of each side of that square?"

In this framework, $a$ is a positive number, and the exponent ($2$), is a natural number. The square root of $a$ is then that side length (a positive number). Similarly, the cube root of $a$ is the length of the side of a cube which has volume $a$ (again, both the cube root and $a$ are positive numbers).

The notation $\sqrt[n]{a}$ reflects this history—unless one has explicitly noted otherwise, $n$ is a natural number, and $a$ is a positive real number. We certainly *can* extend the definition of the radical notation, but my impression is that this is rarely done.

While this is the bit that I am most uncertain about, my understanding is that a more broadly defined real exponential function, i.e. $x \mapsto \exp_a(x)$, comes about with the rise of calculus in the mid-17th century. In this context, we assume that $a > 0$ and that $x$ is a real variable, which allows us to talk about rates of changes which are proportional to the underlying variable (e.g. the rate at which a colony of bacteria grow is proportional to the size of that colony:
$$ \frac{\mathrm{d}P}{\mathrm{d}t} = kP, $$
where $k$ is some intrinsic growth rate).

In this setting, $\exp(x)$ (the natural exponential function) is the unique solution to the initial value problem
$$ \frac{\mathrm{d}y}{\mathrm{d}x} = y, \qquad y(0) = 1. $$
It can be observed that $\exp(x)$ has a lot of the same properties as $\mathrm{e}^x$ (where the former notation indicates the solution to an IVP, and the latter notation indicates "repeated multiplication"), but showing that these are the same requires a little bit of work.

As such, it is probably healthy to use different notation for these two notions.

Logarithms were first developed in the 16th century by John Napier. In modern language and notation, Napier observed that there is a natural isomorphism between the multiplicative group of positive real numbers, and the additive group of all real numbers. As such, if you wanted to multiply two real numbers, it might save you some time to look up the logarithms of those two numbers in a table, add the results, and then take an antilogarithm (find the number in your table of logarithms whose logarithm is your sum).

Addition and two or three table lookups are relatively quick when compared to multiplying to very large or very precise numbers, so books of logarithms proved to be quite useful. The fundamental idea is that a logarithm is a function $f$ which satisfies the functional equation
$$ f(x+y) = f(x)f(y). $$
Napier's tables of logarithms took $f$ to be the common logarithm (that is, the logarithm with base 10), but it turns out that any log will do.

Again, it is possible to show that the exponential function with base $a$ and the logarithmic function with base $a$ are inverse to each other, but this is a later historical development.

### Pedagogy

Thus far, I have claimed that the notations $a^x$, $\sqrt[n]{a}$, $\exp_a(x)$, and $\log_a(x)$ have distinct historical motivation, and were originally understood as representing very distinct notions. However, this does not necessarily mean that we should continue to regard them as distinct, nor that would should not adopt common notation.

Indeed, this brings me to the crux of what I believe this question is about, and to the nut of my answer as a frame challenge. The underlying question is not "Has anyone considered alternative notation?", but rather "Why don't we use and/or *teach* alternative notation?" and, perhaps, "*Should* we use alternative notation?"

An entirely correct, but also useless, answer is that we don't use alternative notation because we don't. Expanding on this a little, mathematical notation is a kind of language which we use to transmit ideas. Like all language, mathematical notation evolves over time, and is the product of human interaction. We don't use alternative notation because (a) the notation we have is sufficiently well understood by "fluent" or "native" speakers of mathematics, and (b) because the community of people who practice mathematics have never felt a need for such notation—it simply hasn't proved useful enough to dispense with the old notation.

In other words, the language of mathematical notation has not evolved a new set of notation for these relationships because the native speakers of that language have not felt a need for it.

Which brings us to the neophyte speakers—the students who might find a "unified" notation "easier to learn".

In principle, an instructor could introduce a new notation and teach that to students. Indeed, I have sometimes been tempted to dispense with $\pi$ and instead use the notation $\tau$ ($=2\pi$) when teaching trigonometry.

However, I think that such an action would ultimately do an incredible disservice to students. The goal of mathematics instruction is (or, at least, should be) to teach students to "do" mathematics as it is currently "done" by professionals in the community. Part of this requires that we teach students to use the language and notation of actual working mathematicians.

As such, students need to be familiar (and even *comfortable*) with exponential notation, logarithmic notation, function notation, radical notation, and so on. They should be taught the subtle distinctions between these different notations, and should understand when and *why* one notation might be preferable over another.

### Epilog

To completely unify the notation, we can start by writing
$$ z = x^y = \exp(y \operatorname{Log}(x) + i2k\pi) \qquad\text{or}\qquad \exp(\operatorname{Log}(z) + i2k\pi) = \exp(y \operatorname{Log}(x)), $$
where we assume that $x,y,z\in\mathbb{C}$, $\operatorname{Log}$ is the principal branch of the complex logarithm, $\exp$ is the complex exponential function, and $k$ is any integer.

In this notation, we get something like^{[1]}
$$ x = \exp\left( \frac{\operatorname{Log}(z) + i2k\pi}{y}\right), $$
and
$$ y = \frac{\operatorname{Log}(z) + i2k\pi}{\operatorname{Log}(x)}. $$

In other words, there is existing notation which already unifies the various notation of exponentiation, roots, and logarithms. It isn't necessarily "pretty", and it isn't appropriate for elementary students, but it already exists.

[1] I will note that I have been a little sloppy in solving for $x$ and $y$ under the assumption that $x$, $y$, and $z$ are complex. What I have written should be fine if $y$ and $z$ are real, but I was not too careful about chasing complex exponents around.