In this diploma thesis we are gradually building an exponential function. We start from the construction of real numbers. At the beginning we define the notion of binary operation, the concept of groups, rings, fields, concept of a ordered field and the concept of field of real numbers. Using the concept of inductive set, we have defined the natural numbers, then the integer and rational numbers. Then we have defined the concept of the sequence as a function. We have proved an important theorem ”The principle of definition by induction” and have used it to define \(a^{n}, \forall {n} \in \mathbb{N}, a \in \mathbb{R}\) and \(a^{m}, \forall {m} \in \mathbb{Z}, a \in \mathbb{R} \backslash \{0 \}\) and then \(a^\frac{m}{n}\) where \(m \in \mathbb{Z} , n \in \mathbb{N} , a > 0\) and we have proved that this is a good definition, ie. that does not depend on the choice of numbers m or n. Finally, we have defined \(a^{x}\) where is \(x \in \mathbb{R} \backslash \mathbb{Q}, a > 0\). We have built an exponential function with base \(a > 0, a \ne{1}\) as \(exp_{a} : \mathbb{R} \to \langle{0, \infty}\rangle\) by \(exp_{a}(x)=a^{x}, \forall x \in \mathbb{R}, a>1\). We have proved uniqueness and existence of the function. Furthermore, we have shown the different features of such a functions such as continuity, bijectivity and finally derivability. We have defined the inverse function with base a as \({(exp_{a})}^{-1} : \langle{0, \infty}\rangle \to \mathbb{R}\) which is called the logarithmic function and we have proved that this function is differentiable.