Is the set of positive real numbers a ring under these operations? Let $L$ be the set of positive real numbers. Two operations are defined:
$$ a \oplus b = ab$$
$$a \times b = a^{\log b}.$$
Is L a ring?
1) a $\oplus$ b = ab, ab $\in$ L.
2) a x b = $ a^{\log b},  a^{\log b} \in L$.
3) addition is commutative
4)1 is the additive identity element
5) associative multiplication
6)Distributive laws
7) Associative multiplication.
8) Is 1 the additive inverse?
 A: You might apply the fact that if $\phi$ is a ring homomorphism (so respecting the addition and multiplication) from a ring $R$ to a set, then its image $\phi(R)$ is again a ring. Now let $\phi$ be the map from the normal ring of reals $\mathbb{R}$ to $\mathbb{R}_{>0}$ with your addition $\oplus$ and multiplication $\otimes$, defined by $\phi(a)=e^{a}$, where $e$ is the base of the natural logarithm. Obviously this is a well-defined bijection. One can easily check that $\phi$ is a ring homomorphism. Observe that 0 is mapped to 1 (the neutral element w.r.t. $\oplus$), and the unit 1 is mapped to $e$ (the neutral element w.r.t. $\otimes$). Hence, the two rings are even isomorphic!
A: You ask:


Is there a zero element in the set of positive real numbers?


What this means in this context is:


Is there a positive real number $a$ such that $a\oplus x=x$ for all positive real numbers $x$?


The defintion of $a\oplus x$ is $ax$, where the latter is ordinary multiplication.  Is there a positive real number $a$ such that $ax=x$ for all positive real numbers $x$?
You write:


I see that L is a ring with the given properties but...


I don't see how you can see it is a ring if you don't know whether there is an additive identity.  I suggest checking every single one of the properties of a ring to verify whether or not it holds.  You can ask for more specific help if you get stuck on any other parts.
