Consider the exponential function $exp_b(x) = b^x$ for $x \in
R$ and the logarithmic function $log_b(x)$ for $x \in
(0,\infty)$.
When $b \ne 1$ , the values of the function $exp_b$ and $log_b$ vary
in predictable ways depending on whether $b>1$ or $0<b<1$.
This is apparent by reviewing the mapping diagrams along with the
graphs in some of our previous examples.
But first we review two key concepts: increasing and
decreasing.
Two examples.
Notice how the graph and mapping diagram visualize the fact that for
exponential and logarithmic functions, if $b >1$ then
$exp_b$ and $log_b$ are increasing functions while if
$0<b<1$ then $exp_b$ and $log_b$ are decreasing functions.
For an exponential function, $exp_b$, with $0<b<1$, when $x>0$
and is very large in magnitude then $exp_b(x) \approx 0$.
With $1<b$ when $x<0$ and is very large in magnitude then $exp_b(x) \approx 0$.
This behavior is described by saying that the function $exp_b$ is
"asymptotic to 0" and in the graph that the line "$y=0$ is a horizontal
line asymptote for the graph of $exp_b$.
Note that $0$ is not in the domain of $\log_b$.
Due to the fact that $\log_b$ and $exp_b$ are inverse functions,
for a logarithmic function, $\log_b$, with $0<b<1$, when $x>0$
and $x \approx 0$ then $\log_b(x) > 0 $ and is very large in
magnitude.
With $1<b$ and $x \approx 0$ then $\log_b(x) < 0 $ and is very large in magnitude.
This behavior is described by saying that the function $\log_b$ is "asymptotic at
0" and in the graph that the line "$x=0$" is a vertical line asymptote
for the graph of $\log_b$.
You can use this next dynamic example to
investigate visually the effects of the base $b$ simultaneously on
the exponential and logarithmic functions whether the functions are
increasing or decreasing in a mapping diagram and a
graph and the asymptotic behavior of each of the functions connected to the nature of the base, $b$.
ExampleELF.DIDA.0Dynamic
Visualization of Increasing and Decreasing and Asymptotes for Exponential and
Logarithmic Functions: Graphs, and Mapping Diagrams