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.

Definition ID 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.ID.1 : $b = 2; f(x) = 2^x$ and $g(x)= log_2(x)$.
Example ELF.ID.2 : $b= \frac 1 2 ; f(x) = (\frac 1 2) ^x$ and $g(x)= log_ {\frac 1 2} (x)$.

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.

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