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.

Definition IDA Increasing/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.
Example ELF.IDA.1 : $b = 3; f(x) = 3^x$ and $g(x)= log_3(x)$.
Example ELF.IDA.2 : $b= \frac 1 2 ; f(x) = (\frac 1 2) ^x$ and $g(x)= log_ {\frac 1 2} (x)$.

Asymptotes for $exp_b$ and $\log_b$
The increasing and decreasing nature of these functions has another important aspect connected their behavior.

This behavior is similar to  some of the behavior studied with rational functions, what is usually described with the term "asymptotic".

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, $b$.

Example ELF.DIDA.0 Dynamic Visualization of Increasing and Decreasing and Asymptotes for Exponential and Logarithmic Functions: Graphs, and Mapping Diagrams