ELF.DOM.L The Domain for Logarithmic Functions
Consider the logarithmic function $f(x) = \log_b(x)$.

The standard domain of a logarithmic function is $\{ x \in  R: x >0\}$.
This is visualized on the graph of $f$ by the fact that the curve appears only on the half plane $H^+ = \{(x,y) : x >0\}$.
For the mapping diagram of $f$ the domain of $f $ is visualized by the fact that arrows for the function emanate only from the open ray in the domain axis: $(0, \infty)$.
When another function, $h$,  is defined by composition of a function $g$ with the logarithm, so $h(x) = f(g(x)) = \log_b(g(x))$  the domain for $h$ is restricted to $\{x: g(x) >0 \}$.
This quality of the logarithm functions in compositions is illustrated in mapping diagrams of the compositions showing how this domain restriction makes sense.
Example  DOM.L.1 : Suppose $f(x) = \log_2(x) $. Visualize the domain of $f$ on a graph and a mapping diagram.

Example DOM.L.2 : Suppose $h(x)= \log_2(3-x)$Visualize the domain of $h$ on a graph and a mapping diagram.
The next example shows an important connection between a mapping diagram for a logarithmic function as a composition with a linear function.
Example DOM.L.3 : Suppose $h(x)= \log_2(2x+5) $. Visualize the domain of $h$ on a mapping diagram.
The included GeoGebra mapping diagram can be used to visualize the domain for other compositions $h(x) = f(g(x)) = \log_b(g(x))$ .
Try $g(x) = |x|$, $g(x) = x^2 -4$ and $g(x) = 4 - x^2$.

You can use this next dynamic example to visualize the domains of logarithmic  like those in Examples DOM.L.1 and DOM.L.2 with a graph and mapping diagram of $f$.

Example ELF.DDOM.L.0 Dynamic visualization of the domains for $h(x) =\log_b(g(x))$ on Graphs and Mapping Diagrams