How to derive that for every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in \mathbb{R}$ such that $y=z^x$. I am not sure on how to derive the following statement concerning the reals (that I think should be true).


  For every real $y &gt; 0$, for every positive real $z \neq 1$, there is a $x \in \mathbb{R}$ such that $y=z^x$.


QUESTIONS:


Is this part of the definition of exponentiation, or it is a result that can be proven?
If it can be proven, how (maybe by using the Archimedean Property)?


Thank you for your time.

PS: Few days ago I asked almost the same question, but a user pointed out (rightly) that the question I really wanted to ask is the one here.
Since the exponential function $\exp(t) = e^t$ strictly increases on $\mathbb{R}$, (Check this in your analysis textbook) its inverse, the logarithmic function $\log t$ on $\mathbb{R}^+$ is well-defined. If we take $x = \log y / \log z$, $ z^x = e^{\log y} = y$ clearly.
$$y = a^{\log_a(y)} \to z=a, x=\log_a (y)$$