Mathematical Operations Substraction is the inverse of addition
Division is the inverse of multiplication
Taking a power is the inverse of taking some root
Having an exponential is the inverse of having a logarithm       
My question is if we can implement any mathematical operational using only one of the remaining operations ( but not the inverse operation ) ?
For example, can we implement addition with only multiplication ? Can we take a logaritm using only subsctraction ?
 A: In any commutative, semigroup $(G,\star)$ you can define the $n$-th power of an element $g\in G$ to be the shorthand $g^n=g\star\dots \star g$ ($n$-times).
Then the $n$-th powers in $(\Bbb Z, +)$ are given by multiplication by $n$ and the $n$-th powers in $(\Bbb Z\setminus\{0\},*)$ are given by exponentiation.
Moreover, if $(G,\star)$ is actually a (commutative) group, then you can define an inverse to
$$
\begin{align*}
(\star g):G&\to G\\
h&\mapsto h\star g
\end{align*}
$$
by using the inverse of $g$ in $G$, i.e. $\bar{g}$ such that $g\star\bar{g}=e$
$$
\begin{align*}
(\star \bar{g}):G&\to G\\
h&\mapsto h\star \bar{g}
\end{align*}
$$
Then in our previous examples $(+(-y))$ is an inverse to (+y) in $(\Bbb Z,+)$ and $(*\frac{1}{y})$ is an inverse to $(*y)$ in $(\Bbb Q\setminus \{0\},*)$, which is the smallest group containing $(\Bbb Z\setminus\{0\},*)$.
Note that it is possible to define partial inverses in a semigroup, too, like subtraction in $\Bbb N$.
Coming to your example question, the answer is no. You cannot reconstruct
$$
\star:G\times G \to G
$$
knowing only the power maps
$$
\begin{align*}
(-)^{(-)}:\Bbb Z \times G &\to G\\
(a,b) &\mapsto b^a
\end{align*}
$$
unless in some special cases, e.g when $(G,\star)$ is a cyclic group, i.e. if there is a $g\in G$ such that every $h\in G$ is of the form $h=g^n$ for some $n\in \Bbb Z$.

To clarify things a bit, suppose we wish to define addition in $\Bbb Z$ using multiplication. By the distributive property we get that for all $a,b,c\in \Bbb Z$
$$
ab+ac=a(b+c)
$$
hence the only possible definition is the trivial one.

As a side note, all this is about algebraic definitions of things, and in particular about defining in $\Bbb Z$ addition through multiplication or multiplication through powers. You get a somewhat different picture from the analytical point of view. For example, on $\Bbb R$ you can define
$$
e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots
$$
