# Logarithmic Function

The logarithmic function is the inverse of the exponential function. Therefore, its domain and range are inverted compared to the exponential function. A logarithmic function is defined as a function of the form:

\[ y = \log{_a}x \quad \text{with} \quad a \gt 0 \quad a \neq 1 \quad \forall x \in \mathbb{R}^+\]

- Domain: \(x \in \mathbb{R}^+ \)
- Range: \(\mathbb{R}\)

We have two cases:

- for \(a \gt 1\) the function in increasing.
- for \(0 \lt a \lt 1\) the function is decreasing.

Graphical representation

of the function \(f(x) = \log_a(x)\)

with \(a > 1\).

Graphical representation

of the function \(f(x) = \log_a(x)\)

with \(0 \lt a \lt 1\).

- Derivative: \[\frac{{d}}{{dx}} \log_a(x) = \frac{1}{x \ln(a)}\]

- Indefinite integral: \[\int \frac{\log(x)}{\log(a)} \ dx = \frac{x \cdot (\log(x)-1)}{\log(a)} + c\]

More in general, considering the function \(y= \log(x)\), the natural logarithm, we have:

- \(\log(x) = \log_e(x)\)
- \(\log(x) = \log(a) \cdot \log_a(x)\)

- Domain: \(x \in \mathbb{R}^+ \)
- Range: \(\mathbb{R}\)
- Roots: \(x=1\)

- Derivative: \[\frac{{d}}{{dx}} \log(x) = \frac{1}{x}\]

- Integral representation: \[\int_{0}^{x} \log(t) \ dt\]

- Indefinite integral: \[\int \log(x)dx = x \cdot( \log(x)-1) + c\]

- Definite integral \[\int_{0}^{1} \log(x) \ dx = -1\]