# Logarithmic functions

The function that assigns to the independent variable $x$ the value of $f (x) =\log_ax$ is called logarithmic function of base $a$, where $a$ is a positive real number other than $1$.

We observe that if we apply the log to the exponential function base $a$ we obtain the identity function $$\log_a(a^x)=x$$ Similarly it is satisfied that $$a^{\log_ax}=x$$ Therefore the exponential and logarithmic functions are inverse functions.

## Graph

As in case of the exponential functions, the graph of the logarithmic functions changes if the base is greater to or smaller than $1$.

Let's see it with $f(x)=\log_2x$ and $h(x)=\displaystyle \log_{\frac{1}{2}}x$.

It is remarkable that the logarithmic functions always go thorugh the point $(1, 0)$ since any number to the power $0$ is $1$.

$f(x)=\log_2x$

$\displaystyle f(x)=\log_{\frac{1}{2}}x$

## Properties

From its graphic representation we observe that the logarithmic function satisfies the following properties:

• Domain: $Dom (f) = (0,+\infty)$
• Image: $Im (f) = \mathbb{R}$
• Bounds: It is not bounded.
• Intersection with the axes: It cuts the horizontal axis at $x = 1$. It does not cut the vertical axis.
• Continuity: It is continuous in its domain.
• Asimptotes: The straight line $x = 0$ is a vertical asimptote.
• If $a> 1$: $\displaystyle \lim_{x \to 0^+} \log_ax=-\infty$ and $\displaystyle \lim_{x \to +\infty} \log_ax=+\infty$
• If $0 <1$: $\displaystyle \lim_{x \to 0^+} \log_ax=+\infty$ and $\displaystyle \lim_{x \to +\infty} \log_ax=-\infty$
• Regularity: It is not periodic.
• Symmetries: It is not symmetric.
• Monotonicity: If $a> 1$, the function is strictly increasing. If $a<1$, the function is strictly decreasing.
• Relative extrema: It does not have any.
• Injectivity and exhaustivity: It is injective (the images of different points are different), and also it is exhaustive since the image is $\mathbb{R}$