5.3 Logarithmic Functions
Thursday, December 09, 2010
1:35 PM
Solve this equation:
Now try
What about
(**)
Now notice this, when we defined exponential functions, we had this
This is an exponential with base 3,
So what are we saying about
in the statement above (**)?
and
So when we want to solve for in
That is the property of _(Inverses)__
we can use a property that we defined in Chapter 4.
Where we know that if we could find the inverse of
knew our
that we could find , when we
So, today we talk about the inverse of an exponential function, that is the logarithmic function.
Let's make a table of what the logarithmic function for a given base has to do.
f
-2
1/4
-1
1/2
0
1
1
2
2
4
Notes Page 1
Definition: Logarithmic Function
For b>0, b 1,
The inverse of
is denoted as
, is the logarithmic function with base
Properties of Graphs of Logarithmic Functions
Let
be a logarithmic function, b>0, b
1. Is continuous on its domain (0,
. Then the graph of
:
2. Has no
3. Pass es through point ______
4. Lies to the ________ of the ___ axis, which is a ______________ asymptote.
5. Is ______________ as
increases if ______; is _____________ as
increases if 0<b<1
6. Intersects any horizontal line exactly once, so is __________________
One major skill to have is the ability to convert back and fourth between logarithmic and
exponential form
They are related as follows:
Logarithmic Form Exponential Form
These two statements are equivalent, just two different ways of expressing the same thing.
Let's convert some exponentials into logarithms.
I will do one, you will do three.
----
Now let's do it the other way:
Notes Page 2
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Solutions of the Equation
Find
as indicated
Find
Now you:
Find
Find
Now you:
Find
Find
Now you:
Find
There are quite a few properties for working with logarithms, and they are listed on page 358,
as part of your homework, you will work with these properties.
Notes Page 3
as part of your homework, you will work with these properties.
Notes Page 4