Sunday, November 13, 2011
5.5 Exponential and Logarithmic Equations
Hey Guys! Its bout to be a party
%20=%20log8 "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
 "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
=%20log(2^{1-3x}) "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
log3%20=%20(1-3x)log2 "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
=%20-(log2%20+%20log3%20^{4}) "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
}%20{(log3+%20log8)} "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
}%20{log24} "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
When you are solving an equation with variables as exponents and constants as the base we can "take the log of both sides" to make it easier to solve. For ex:
by then using a calculator you would come to the answer
natural logarithms could have also have been used to obtain
These are both alternatives to using logarithmic form or "the key to everything" as in 5.3
a quick recap
using this method you would have come to the equation
most calculators do not have a 
key so you would then be left with this as your simplified answer.
This is why the new "take the log of both sides" technique is useful
Which leads to the next topic. When dealing with an equation that has a logarithmic base other than e or 10 you can use this handy theorem called the: Change of Base Formula
If
and if a and b are pos real numbers different from 1, then
When using this Formula REMEMBER
AND
ARE NOT TRUE
Also as a refresher
And
Now that we know the Change of Base Formula We can apply it to problems such as the first example.
Using "The Key to Everything" you get
APPLY
to get
Just as before
A little more practice
you can also use it in exponential equations
Thursday, November 10, 2011
5.4 Properties Of Logarithms
Laws of Logorithms
Log a (uw) =log a (u) + log a (w)
Log a (u/w)= log a (u) - log a (w)
Log a (u^c) = c log a u. For every real number c
But remember, logarithms can be sneaky, and aren't always what you think. For example
Log a (u+w) Does NOT = log a u + log a w
Also
Log a (u-w) does NOT = log a u- log a w
How to use the laws of Logorithms
There are two ways to express Logorithms. One is as one Logorithm and the other is in terms of variables.
In order to change from one Logorithm into variables you would take the exponent of the variable and make it the coefficient in front of log.
For example you would change
log a x^3 into 3 log a x.
Remember, the whole point is to use the three laws of Logorithms.
The first step in changing more complex problems is to make sure that you have taken out all the multiplication and division, and replaced it with addition and subtraction.
The next step is to change the exponents to the coefficient, as we did earlier.
Now, for expression as one Logorithm.
The first step is just the inverse of last time. Here we change
3 log a x into log a x^3.
Next we will do the opposite again. Changing. From addition and subtraction into multiplication and division.
Next we solve logarithmic equations.
Log a (2x+3)= log a 11+ log a 3
Now remember- Log a (uw) =log a (u) + log a (w)
I told you Logorithms are sneaky
Because of law one the next step is
2x+3=33
So
X=15
How to solve ane equation like
Log 2 (x+2)= 4
Since you can't just get rid of the log 2's you have to do something with them.
It is important to remember that log 2 of 4 = 2^4. All you have to do is change it to exponential form.
So
X+2= 2^4
X+2=16
X=14
Graphs of logarithmic equations
When graphing log a x^2 it is the same except is reflected about the y axis, in addition to being where it is now.
Matthew "Seaglass" Silbergleit
Wednesday, November 9, 2011
Hey so after some great strategy by Hannah and I, I have yet again gotten the blog post. Today in class we learned about Logarithms.Oh and thank you Mr.Wilhelm for finally changing the background...
In an equation...
we learned that x is being expientiated and to solve this you need to log it.
In this chapter we are learning about exponential functions, and long and behold they are one-to-one. And what do we know about one-to-one functions??? They all have inverses!!! The inverse to exponential functions are logarithmic functions.
We know this because inverse functions are reflected over X=Y.
Now the KEY TO EVERYTHING (also called the Definition of loga in the book) is that...
The first equation is in exponential form when the second one is in logarithmic form.
Changing equations from exponential form to logarithmic form and vice versa is fairly easy. All you have to do is this...
(Index means exponent)
Now to solve logarithmic equations is pretty easy too. The easiest way to do it is to change it to exponential form and we already know how to solve those.
Given
Change to exponential form
Simplify
add 4 to both sides
Now there are two different types of logs.
The common log
The natural log (the one we are gonna use most)
In the homework there were equations that looked like this
which are far less confusing than you would think.
All this is is another way to write exponential functions.
This just means that..
or
Finally we learned about the graph of logarithmic functions.
Changes-
a-vertical stretch
b-higher number makes the graph grow slower
c-Shifts left/right
d- shifts up/down
So that's pretty much it for logarithms. Thanks again to Hannah for being a very sneaky and loyal friend.
And everyone remember that finials are approaching and there is a good chance that we shall all fail.
-Jennifer Kendall
In an equation...
we learned that x is being expientiated and to solve this you need to log it. In this chapter we are learning about exponential functions, and long and behold they are one-to-one. And what do we know about one-to-one functions??? They all have inverses!!! The inverse to exponential functions are logarithmic functions.
Now the KEY TO EVERYTHING (also called the Definition of loga in the book) is that...
Changing equations from exponential form to logarithmic form and vice versa is fairly easy. All you have to do is this...
(Index means exponent)
Now to solve logarithmic equations is pretty easy too. The easiest way to do it is to change it to exponential form and we already know how to solve those.
Given Change to exponential form Simplify
add 4 to both sides Now there are two different types of logs.
The common log The natural log (the one we are gonna use most) In the homework there were equations that looked like this
which are far less confusing than you would think. All this is is another way to write exponential functions.
This just means that..
or
Finally we learned about the graph of logarithmic functions.
a-vertical stretch
b-higher number makes the graph grow slower
c-Shifts left/right
d- shifts up/down
So that's pretty much it for logarithms. Thanks again to Hannah for being a very sneaky and loyal friend.
And everyone remember that finials are approaching and there is a good chance that we shall all fail.
-Jennifer Kendall