1 Introduction
2 Mathematics is not only about computing
3 Basic concepts
4 Elementary functions
4.1 What is it a function?
4.2 The absolute value
4.3 Lower and upper integer part
4.4 Linear function
4.5 Quadratic function
4.6 Polynomial function
4.7 Roots
4.8 Rational function
4.9 Trigonometric functions
4.10 Exponantiation and logarithm
5 Analytical geometry
6 Warning
7 List of the used symbols
Index
Bibliography

For $0 < a \neq 1$ the function^{45}

\begin{equation*}
f(x) = a^x, \quad x\in D_f = \R,
\end{equation*}

is called **exponentiation of base $a$**. This function extends the operation of elevation to power
to non-integer exponents. For any real numbers $x$ and $y$, it applies the well known equality

\begin{equation*}
a^x \cdot a^y = a^{x+y} \quad \text{a} \quad \big(a^x\big)^y = a^{xy}.
\end{equation*}

In figure 4.14 the graph of the function $f$ is shown for different bases $a$.

In general, for $a > 1$ $f$ is strictly increasing (i.e. $f(x) < f(y)$ for any $x < y$), $D_f = \R$ and $H_f = (0,+\infty)$. For $a < 1$, $f$ is strictly decreasing (i.e. $f(x) > f(y)$ for any $x < y$), $D_f = \R$ and $H_f = (0,+\infty)$.

The **logarithm** is the inverse function of the exponentiation (only
in the case of base different from one, otherwise the exponential function is not injective).
More specifically, from the graph of the exponential function $f(x) = a^x$, $a \neq 1$, we see that
for every real number $y$ there exists a real $x$ such that $a^x = y$. We say that a function with such a property is injective (in this case on the whole $\R$) and therefore invertible on its image set.The inverse function of the exponentiation of base $a$, $0 < a \neq 1$, is said **logarithm of base $a$** and we denote it as $\log_a$. The domain of the exponentiation is the
whole $\R$ and its image set is the interval $(0,+\infty)$.
From this it follows that the domain of the logarithm, since it the function inverse of the exponentiation, is $D_{\log_a} = (0,+\infty)$ and
its image set is $H_{\log_a} = \R$.

The reader has certainly already indirectly encountered logarithms through applications. For example the Richter scale (that measures the intensity of earthquakes) or the decibel scale (measuring the intensity of sound) are logarithmic.

Important properties of the logarithm can be derived from properties of the exponentiation:

\begin{equation}\label{eq-log-1}\tag{4.16}
a^{\log_a x} = x, \quad x>0,
\end{equation}

\begin{equation}\label{eq-log-2}\tag{4.17}
\log_a a^x = x, \quad x\in\R,
\end{equation}

\begin{equation}\label{eq-log-3}\tag{4.18}
\log_a xy = \log_a x + \log_a y, \quad x,y > 0,
\end{equation}

\begin{equation}\label{eq-log-4}\tag{4.19}
\log_a x^y = y \log_a x, \quad x>0 \ \text{a} \ y\in\R.
\end{equation}

Indeed, the first two equalities, (4.16) and (4.17), are merely an expression of the inverse relationship between the exponential and the logarithm, thus they apply by definition. Let us prove the equality (4.18). For positive $x,y$ there exist real $u,v$ such that

\begin{equation*}
x = a^u \quad \text{a} \quad y = a^v.
\end{equation*}

From this we have

\begin{equation*}
xy = a^u \cdot a^v = a^{u+v}.
\end{equation*}

Thus

\begin{equation*}
\log_a xy = u + v = \log_a x + \log_a y.
\end{equation*}

In a similar way, the property (4.19) can be proven.

Remark 4.3

The reader is certainly familiar with the operation called *remove the logarithm*.
That is, saying the following: if

\begin{equation*}
\log_a x = \log_a y,
\end{equation*}

for some $x,y > 0$ and $0 < a \neq 1$, then

\begin{equation*}
x = y.
\end{equation*}

This operation is no magic. It is just about using the injectivity of the function $\log_a$. The same can be done with any injective function!

Question
4.11