The inverse operations
In addition to the direct operations of addition, multiplication, and raising to a power, we have also the inverse operations, which are defined as follows. Let us assume that $a$ and $c$ are given, and that we wish to find what values of $b$ satisfy such equations as $a + b = c$, $ab = c$, $b^a = c$. If $a + b= c$, $b$ is defined as $c - a$, which is called subtraction. The operation called division is also clear: if $ab = c$, then $b = c/a$ defines division—a solution of the equation $ab = c$ “backwards.” Now if we have a power $b^a = c$ and we ask ourselves, “What is $b$?,” it is called the $a$th root of $c$: $b = \sqrt[a]{c}$. For instance, if we ask ourselves the following question, “What integer, raised to the third power, equals $8$?,” then the answer is called the cube root of $8$; it is $2$. Because $b^a$ and $a^b$ are not equal, there are two inverse problems associated with powers, and the other inverse problem would be, “To what power must we raise $2$ to get $8$?”  This is called taking the logarithm. If $a^b = c$, we write $b = \log_ac$. The fact that it has a cumbersome notation relative to the others does not mean that it is any less elementary, at least applied to integers, than the other processes. Although logarithms come late in an algebra class, in practice they are, of course, just as simple as roots; they are just a different kind of solution of an algebraic equation. The direct and inverse operations are summarized as follows: \begin{equation} \begin{alignedat}{5} &(\text{a})&&\quad \text{addition}&&\quad &&(\text{a}')&&\quad \text{subtraction}\\ & &&\quad a+b=c&&\quad && &&\quad b=c-a\\ &(\text{b})&&\quad \text{multiplication}&&\quad &&(\text{b}')&&\quad \text{division}\\ & &&\quad ab=c&&\quad && &&\quad b=c/a\\ &(\text{c})&&\quad \text{power}&&\quad &&(\text{c}')&&\quad \text{root}\\ & &&\quad b^a=c&&\quad && &&\quad b=\sqrt[a]{c}\\ &(\text{d})&&\quad \text{power}&&\quad &&(\text{d}')&&\quad \text{logarithm}\\ & &&\quad a^b=c&&\quad && &&\quad b=\log_ac\\ \end{alignedat} \label{Eq:I:22:2} \end{equation}  Now here is the idea. These relationships, or rules, are correct for integers, since they follow from the definitions of addition, multiplication, and raising to a power. We are going to discuss whether or not we can broaden the class of objects which $a$, $b$, and $c$ represent so that they will obey these same rules, although the processes for $a + b$, and so on, will not be definable in terms of the direct action of adding $1$, for instance, or successive multiplications by integers.