How do I balance an equation that contains a variable nth root? For example, this is easy enough to solve for x (given that y and z are known):
    x^y = z
    x = y√z
    As in x^8 = 256, x = 8√256, x = 2.

I know that to balance a root you apply a power, and vice versa.
But when the power is the variable, you end up with an unknown nth root to balance.
    y^x = z
    y = x√z
    x = ?
    As in 2^x = 256, 2 = x√256, ..., x = 8.

I'm trying to figure out the required step noted by the ellipsis. I feel like I'm missing something simple.
Update:
After messing around a bit, I was also able to plug the equation into wolframalpha.com as: solve for x where 2 = 256^(1/x) (the last part being exponent notation; Wolfram doesn't understand xrty(x,y), root(x)(y), or x√y). I get the correct answer, but need to be a Pro member to see the steps. Either way, they do reveal that x√y = y^(1/x), and that logarithms are involved, as confirmed in the answers below.
 A: The easiest way to compute the value of unknown exponents is to take the logarithm on both sides. For example, if $2^x = 256$ then $x\log(2) = \log(256) \implies x = 
\log(256)/\log(2) = 8$.
A: Let me use some better notation. Let $a$ and $b$ be given, and let $x$ be a variable. (I will assume we are working in the real numbers $\mathbb{R}$ for the duration of this answer.) How do you solve $\displaystyle b^x=a?$ This question directly motivates the definition of the logarithm function.
Definition. The base $b$ logarithm of $a$, denoted $\log_b a$, is the real number $x$ such that $b^x=a$.
The number $x$ is unique for a given $a$ and $b$, because $\log_b$ is a bonafide function. Before answering your question, let us look at a numerical example. 
Example. In terms of logarithms, what $x$ satisfies the equation $2^x=1024$? By the above definition, we see $\log_2(1024)=x$ is the correct answer. Computing the first few powers of $2$, we see $x=10$ is the solution. 
Because $\log x$ is such a nice, fundamental function—indeed, it is the inverse of "the most important function in mathematics" $\exp x$ (Rudin)—it is included on calculators. Decades ago, students instead used slide rules or logarithm tables, which is one reason so much time is spent in modern classes on reducing fractions and changing logarithm bases. 
To understand the answer to the question, we need the following fundamental property of logarithms: $\log_b(x^y)=y\log_b(x)$. We also need to note that the notation $\log x$ usually means $\log_e x$, where $e$ is Euler's number, but oftentimes high school courses use $\log x$ to mean $\log_{10} x$, preferring $\ln(x)$ (the "natural logarithm") for base $e$. For the argument below, the base is actually unimportant, so feel free to imagine it being either $e$ or $10$. 
Now that you know what a logarithm is you should be able to understand the following algebra: 
If $b^x=a$, then $\log b^x = \log a$, because $\log$ is a function and returns inputting the same argument in a different representation does not change the result. Then, by the logarithmic power rule, we have $x\log b=\log a$, and finally, we divide through by $\log b$ to get 
$$\boxed{x=\frac{\log a}{\log b}}.$$
Nobody computes logarithms by hand anymore, so for given $a$ and $b$, you can simply use a calculator. For example, if $2^x=1073741824$, then $x=\log(1073741824)/\log(2)=30.$ This works for non-integer cases, too. 
I encourage you to look up other basic facts of the logarithm function and their proofs. The beginning parts of the Wikipedia article seem like a decent place to start. Khan Academy has videos, too. 
References.
Rudin, Walter, Real and complex analysis., New York, NY: McGraw-Hill. xiv, 416 p. (1987). ZBL0925.00005.)
