Logarithm Properties
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Complete the following formulas.

1. ** **

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2. ** **

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3. ** **

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4. ** **

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5. ** **

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**THERE IS NO SUCH PROPERTY FOR SUMS OR DIFFERENCES!!!!!**

6. ** **

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** **

**THERE IS NO SUCH PROPERTY FOR SUMS OR DIFFERENCES!!!!!**

7. ** **

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Note in this case the exponent
needs to be on the WHOLE argument of the logarithm. For instance,

However,

8. Write down the change of base
formula for logarithms.

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Write down the change of base
formula for logarithms.

This is the most general change of
base formula and will convert from base *b*
to base *a*. However, the usual reason for using the
change of base formula is so you can compute the value of a logarithm that is
in a base that you can’t easily compute.
Using the change of base formula means that you can write the logarithm
in terms of a logarithm that you can compute.
The two most common change of base formulas are

In fact, often you will see one or
the other listed as THE change of base formula!

In the problems in the Basic Logarithm Functions section you computed the
value of a few logarithms, but you could do these without the change of base
formula because all the arguments could be wrote in terms of the base to a
power. For instance,

However, this only works because
49 can be written as a power of 7! We
would need the change of base formula to compute .

OR

So, it doesn’t matter which we
use, you will get the same answer regardless.

Note as well that we could use the
change of base formula on ** **if we wanted to as well.

This is a lot of work however, and
is probably not the best way to deal with this.

9. What
is the domain of a logarithm?

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10. Sketch the graph of and .

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Not much to this other than to use
a calculator to evaluate these at a few points and then make the sketch. Here is the sketch.

From this graph we can see the
following behaviors of each graph.

Remember that we require in each logarithm.