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 the 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.
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.
ExampleELF.DID.0Dynamic
Visualization of Increasing and Decreasing for Exponential and
Logarithmic Functions: Graphs, and Mapping Diagrams