Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$ Conjecture:

There does not exist a non-integer $x$ such that
$$2^x=a$$ $$3^x=b$$ $$5^x=c$$
where $a,b,c$ are all integers.


I'm aware that the similar question 

There does not exist a non-integer $y$ such that
$$2^y=A$$ $$3^y=B$$
where $A,B$ are all integers.

is a famous unsolved problem.
(evidence in the corrolaries here or here)
My idea was that the addition of the condition $5^x$ made the problem easier and therefore solvable.

Theorem of Siegel that for real $\lambda$ and distinct primes $p,q,r,$ the numbers $$  p^\lambda, \;   q^\lambda, \;   r^\lambda,   $$
cannot all be rational unless $\lambda$ is an integer.
See page 455 in Alaoglu and Erdos, On Highly Composite and Similar Numbers (1944), also chapter 2 of Introduction to Transcendental Numbers by S. Lang.
I see there is also a book Transcendental Numbers by our hero, C.L. Siegel, that could easily give the first polished discussion of the result, which is just what we call a "personal communication" in the Alaoglu Erdos article.
I do not have the Lang book from the library yet. Apparently this covers the same material, and is certainly from the same time: LANG PDF 
From the book, now clear how it relates, see Corollary 1, with the definition of multiplicatively independent middle of page 8 : 


This (Theorem 1, I guess) is this : http://en.wikipedia.org/wiki/Six_exponentials_theorem
