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NARRATOR: Exercise 19.

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Consider the graph
y equals log base 3 of x.

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Explain why the following
two transformations of the graph

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have the same effect.

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A dilation in the x-direction
from the y-axis with a factor of 9,

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or a translation by 2 units down,
that is in the negative y-direction.

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Relate your answer to exercise 16.

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Consider the two transformations.

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A dilation in the x-direction from
the y-axis with a factor of 9...

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..is a result of multiplying 
all of the x coordinates by 9.

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And applying this transformation

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to the function
y equals log base 3 of x,

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we get the equation y equals
log base 3 of x divided by 9.

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Applying the translation
two units down,

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we get the final equation
y equals log base 3 of x minus 2.

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So a dilation of log base 3 of x

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in the x-direction from the y-axis
with factor 9

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results in the equation y equals
log base 3 of x divided by 9.

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Using log laws, we can split this
into two separate logs -

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log base 3 of x minus
log base 3 of 9.

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And we note that 9 is in fact
equal to 3 squared.

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Again, using another log law,
log base 3 of 3 squared

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is equivalent to
2 times log base 3 of 3.

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And then we know that log base 3 of 3
in fact equals 1.

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So we find that
log base 3 of x divided by 9

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is in fact the same

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as log base 3 of x minus 2, which was

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the equation that we obtained after

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a translation by two units down.

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So therefore, for the rule
y equals log base 3 of x,

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a dilation by a factor
of 9 from the y-axis

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has the same effect as
a translation by two units down.

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Lastly we were asked to relate
our answer to exercise 16.

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And exercise 16 looks at the graph
of y equals 3 to the power of x

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and asks you to explain

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why the following two transformations
of this graph have the same effect.

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That is a dilation in the y-direction
from the x-axis with a factor of 9

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and a translation
by 2 units to the left,

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that is in the negative x-direction.

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So the relationship here
is clearly to do with inverses.

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Y equals 3 to the power of x and y
equals log base 3 of x are inverses.

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So if we were to dilate the log graph
by a factor of 9 from the x-axis

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and dilate the exponential graph
by a factor of 9 from the y-axis,

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we will end up with two graphs
that are in fact still inverses.

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And similarly, if we translate
the log graph by two units down

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and translate the exponential graph
by two units to the left,

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we will also result in
two graphs that are inverses.

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And as we've already shown,
the result of these translations

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is the same as the result
of the dilations.

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So if we have the two graphs

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y equals 3 to the power of x and y
equals log base 3 to the power of x,

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the exponential
being from exercise 16

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and the log being from exercise 19,

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we see in fact
that they are inverses,

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that is they are reflections of
each other in the line y equals x.

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And if we also look at the graphs
after the dilation...

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In the case of the exponential,

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a dilation from the x-axis
by a factor of 9,

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and in the case of the log,

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a dilation from the y-axis
by a factor of 9,

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we see that we still end up
with inverse functions.

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That is, graphs that are
reflections of each other

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in the line y equals x.