Prove that I can always write a number a, a>0 as any number c, c>0 to the power of some number (a=c^x)

Question

I'm very new to math, I'm sorry if my question is stupid. I started to study math by my own so I can study Computer Engeneering.

I'm studying logarithms and I try to come up with simple proofs of the properties I learn as often as possible. In trying to make sense of the change of base formula, I came up with the following reasoning:

The change of base formula is $\log_ab=\log_cb/\log_ca$. Assuming I can always write a number $a, a>0$ as any number $c, c>0$ to the power of some number ($a=c^x$):

$$y=\log_ab=\frac{\log_c((c^x)^y)}{\log_c(c^x)}=\frac{xy}x=y$$

I've seen other -and better- proofs, but that is the one I thought. The question is: how can I be sure that my assumption is true?

Sorry for any mistakes, English is not my native language.

Answer 1

Hint
What happens if you chose $a=0$ and $c\ne 0$?

Edited Question
With some further restrictions, this is indeed true. If both $a$ and $c$ are positive real numbers then there exists a unique real number $x$ such that $$a=c^x$$ This number is defined as $$x = \log_c a$$ This also means that any proof of this fact cannot use the logarithm since that would create a tautology.

Answer 2

If $c>1$, then the function $x\mapsto c^x$ is continuous, so it takes all values between its limit in $-\infty$, which is $0$, and its limit in $+\infty$, which is $+\infty$.

If $0<c<1$ then $c^x=\left(\displaystyle\frac1c\right)^{-x}$, so again $c^x$ takes all positive real values.