Martin Flashman, Jan. 10, 2014. Created with GeoGebra

Notice how the arrows on the mapping diagrams are paired with
the points on the graph of the functions.

You can move the point for $x$ on the mapping diagram to see how
the function value for the function $f(x)=\exp_{\frac 1 2}(x)
= (\frac 1 2) ^x$ changes both on the diagram and on the
graph.

Notice how the graph and mapping diagram visualize the fact that for
exponential and logarithmic functions if $ 0<b<1$ then $\exp_b $ and $\log_b$ are decreasing functions.

Uncheck the upper box to hide the graph and arrows on the diagram
for the function $f(x) =\exp_{\frac 1 2}(x) = (\frac 1 2)
^x$.

Check the middle box to show the graph and arrows on the diagram
for the function $g(x) = \log_ {\frac 1 2} (x)$.

Leave the upper box checked to see some of the symmetry between
the graphs and mapping diagrams of these two functions.

Check the lower box to show the confirmation that the paired data
do satisfy the equation: $\exp_{\frac 1 2}(\log_ {\frac 1 2} (y))
= y $ for all $y \in (0,\infty)$.