You can move the points for $x$ and $y$ on the mapping diagrams to
see how
the function values for the functions $f(x) =3^{x-2} = \frac
{3^x}{ 3^2}$
and $g(y) = \log_3(\frac y9) = log_3(y) - log_3(y)$ change on the
diagrams with the different procedures applied in the prescribed
orders.

You can change the base as well as the values of $a$ and $b$ using the sliders.

$\exp_3(x \cdot 2) = (\exp_3(x))^2 = 3^{x\cdot 2} = (3^x)^2$ |

$\log_3(y^2) = 2 \cdot log_3(y)$ |

Martin Flashman, 01 Sept 2016, Created with GeoGebra