7-5 Logarithmic Functions!

Hey guys! Happy Valentine's Day!!!




So here is lesson 7-5 Logarithmic Functions:

Multiply-Add Property of Logarithmic Functions:

• Logarithmic Functions are invereses of exponential functions. With exponential functions the pattern is add-multiply so with Logarithmic functions the pattern is MULTIPLY-ADD.
  
  Example:
  
  
  
  In the above table for a logarithmic function you can see the Multiply-Add pattern. When you multiply the x's by 3 you add one to the y's.
  
  Particular Equation for Logarithmic Functions:
  
  

The General Form for a Logarithmic Function is: y=a+b(logx) or y=a +b(lnx)

• When solving for a and b you plug in two sets of points into the general form similarily to solving for exponential functions. Then, instead of dividing the two equations you subtract one from the other. Then you solve for b and then you plug in the b value into one of the original equations and solve for a. When you have both a and b you can put them back into the general form and you have your equation.
  

Example Problem:

Taking the points (6,1) and (18, 2) from the above table plug them into the general form for x and y to get:

1= a +b(ln6)

and

2= a + b(ln18)

Then subtract the second equation from the first to get :

-1=b(ln6)-b(ln18)

Next you can factor out the b to get:

-1=b(ln6-ln18) --> -1= b(ln 6/18)

b= -1/ln(1/3) which = .910...

Now that we have the b value we can plug it back into either of the original equations for a.

1= a =+.910(ln6)

a=-.631

Now when we plug in the a and b values we get y=-.631 + .910(lnx)

Graphs of Logarithmic Functions:

The Graph of natural log y=ln(x) goes through (1,0) and looks like this:

Now, if we were asked to graph y= 3ln(x-1) we know what it is the above graph with a vertical dilation of 3 and a horizontal translation of 1 (to the right). It would look like this:

Now, the graph of y= -ln(x-3) would be different. Because of the negative sign the graph would reflect over the x axis and it would also shift 3 to the right looking like this:

The graph of log(x^2-1) would look like this (it is reflected around the origin because it is squared):

The domain of this graph is x<-1 or x >1.

For More Help with the graphs go to: http://www.sosmath.com/algebra/logs/log4/log42/log422/log422.html

For help with the equations go to: http://www.sosmath.com/algebra/logs/log4/log47/log47.html

So here is a link to a clip from the movie White Nights where Mikhail Baryshnikov(one of my favorite dancers EVER) does 11 pirouettes. He is amazing and his turns and jumps are crazy!!

http://youtube.com/watch?v=VHbvHoAWKV0

Reminder: Natalie you're next!