NARRATOR: Exercise 19.
Consider the graph y equals log base 3 of x.
Explain why the following two transformations of the graph
have the same effect.
A dilation in the x-direction from the y-axis with a factor of 9,
or a translation by 2 units down, that is in the negative y-direction.
Relate your answer to exercise 16.
Consider the two transformations.
A dilation in the x-direction from the y-axis with a factor of 9...
..is a result of multiplying all of the x coordinates by 9.
And applying this transformation
to the function y equals log base 3 of x,
we get the equation y equals log base 3 of x divided by 9.
Applying the translation two units down,
we get the final equation y equals log base 3 of x minus 2.
So a dilation of log base 3 of x
in the x-direction from the y-axis with factor 9
results in the equation y equals log base 3 of x divided by 9.
Using log laws, we can split this into two separate logs -
log base 3 of x minus log base 3 of 9.
And we note that 9 is in fact equal to 3 squared.
Again, using another log law, log base 3 of 3 squared
is equivalent to 2 times log base 3 of 3.
And then we know that log base 3 of 3 in fact equals 1.
So we find that log base 3 of x divided by 9
is in fact the same
as log base 3 of x minus 2, which was
the equation that we obtained after
a translation by two units down.
So therefore, for the rule y equals log base 3 of x,
a dilation by a factor of 9 from the y-axis
has the same effect as a translation by two units down.
Lastly we were asked to relate our answer to exercise 16.
And exercise 16 looks at the graph of y equals 3 to the power of x
and asks you to explain
why the following two transformations of this graph have the same effect.
That is a dilation in the y-direction from the x-axis with a factor of 9
and a translation by 2 units to the left,
that is in the negative x-direction.
So the relationship here is clearly to do with inverses.
Y equals 3 to the power of x and y equals log base 3 of x are inverses.
So if we were to dilate the log graph by a factor of 9 from the x-axis
and dilate the exponential graph by a factor of 9 from the y-axis,
we will end up with two graphs that are in fact still inverses.
And similarly, if we translate the log graph by two units down
and translate the exponential graph by two units to the left,
we will also result in two graphs that are inverses.
And as we've already shown, the result of these translations
is the same as the result of the dilations.
So if we have the two graphs
y equals 3 to the power of x and y equals log base 3 to the power of x,
the exponential being from exercise 16
and the log being from exercise 19,
we see in fact that they are inverses,
that is they are reflections of each other in the line y equals x.
And if we also look at the graphs after the dilation...
In the case of the exponential,
a dilation from the x-axis by a factor of 9,
and in the case of the log,
a dilation from the y-axis by a factor of 9,
we see that we still end up with inverse functions.
That is, graphs that are reflections of each other
in the line y equals x.