If $a$ is the given base, $a>0$, $a \neq 1$, and if $x$ is any real number, then the function

$$f(x) = a^x$$

is called **the exponential function**.

The base $a = 10^x$ has the special role in the calculating of powers. Let’s observe powers of the shape $10^x$.

For this purpose, in the coordinate plane we will draw the graph of the function $f(x) = 10^x$.

Therefore, the graph of the function $f(x) = 10^x$ is:

As we can see, for the positive real numbers $x$, this exponential function grows very fast, and for the negative real numbers $x$ the function falls to the zero and it’s very close to the negative part of the $x$ – axis.

Similarly, we can draw the graph of the function $f(x) = a ^x$, for any base $a$, $a>0$, $a \neq 1$. For instance, we will draw the graph of the function $f(x) = 3^x$:

Now we will observe the exponential function with the base $0 < a < 1$. For instance, let be $a = \displaystyle{\frac{1}{3}}$. In the same coordinate plane we will draw functions $f(x) = 3^x$ and $g(x) = \left( \displaystyle{\frac{1}{3}} \right)^x$:

We can notice that functions $f$ and $g$ have the same values for numbers of opposite signs, because $f(-x) = 3^{-x} = g(x)$. Therefore, graphs of these functions are symmetrical to the respect of the $y$- axis.

**Properties of exponential function**

The exponential function $f(x) = a^x$, $a>0$, $a \neq 1$, has the following properties:

- The function is defined for every real number $x$.
- All values of the function are positive real numbers.
- $a^x \cdot a^y = a^{x+y}$
- $(a^x)^y = a^{xy}$
- $(a \cdot b)^x = a^x \cdot b^x$
- $a^0 =1$
- If $a>1$, then for $x_1 < x_2$ is valid $a^{x_1} < a^{x_2}$
- If $0<a<1$, then for $x_1 > x_2$ is valid $a^{x_1}> a^{x_2}$