Wednesday, May 18, 2011

Logarithms to Bases other than 10!

Today's class was all about logs.

Unfortunately, Instead of learning about tress and the beauty of nature we spend our time discussing Logarithms to Bases other than ten; which look a little bit more like this...

We started our class by going over homework from the previous night, 9-5, which discussed the basic ideas of logs. Just a quick review:
A common logarithm allows you to solve for x when 10^x=a.The formal definition of Logarithm of x to the base of 10 states that Y is the logarithm of x to the base o f10, or the lo off x to the base of 10, or the log base 10 of x, written y=log3 of x, if and only if 10^y=x
For example:

2=log10 of 100, because 10^2 =100.
The idea of logarithms was carried over to today's lesson, however, it answers the question "what happens if the base is not 10?" Since I know that all of you are just dying to know the answer to that, i shall now proceed and explain everything step by step.

The formal definition of Logarithm of a to the base b states that if b>0 and b is unequal to 1, then x is the logarithm of a to the base b, written x=log b of a, if and only if b^x=a
For example, because 3^5=243, you can write 5=log3 of 243. This
is read "5 is the logarithm of 243 to the base 3" or "5 is the log base 3 of 243".

These are examples of powers which have been converted from the exponential, to the logarithmic form.
3^4 = 81
log3 of 81=4
log3 of 27=3
3^2 =9
log3 of 9=2
log3 of 1=0
3^-2= 1/9
log3 of 1/9 =-2

In order to change a logarithm from the exponent to log form first the equation must be rewritten in the b^n=m form.
For example,
P= (91.028)^X
Next, divide both sides by 9
1.028^x= P/9
The base is 1.028

log1.028 of (P/9)=X

Here are examples of how to solve a logarithm to base other than 10:
Let log 7 of 49= X
  1. 7^X=49 Definition of logarithm
  2. 7^X=7^2 Rewrite 40 as a power of 7
  3. X= ? Equate the exponents
So, log 7 of 49=2

More examples from today's notes:

log10 of 100=2

log2 of 128= 7

log8 of (1/8)= -1

log9 of 3= (1/2)

log7 of 7^4/3= 4/3


Both the exponential equation 3^y=X and the log equation y=log3 of X describe the inverse of the function with equation y=3^x. In general the logarithm function with base b,g(x0=log b of x, is the inverse of the exponential function with a base b,f(x)=b^X
It is important to remember the restrictions of the graphs.
Domain: all reals
For the function y=10^x
Range: y>0
X int: none
Y int: 1
Asymptote: x axis

For the function y=10^x
Domain: x>0
Range: all reals
X int: 1
Y int: none
Asymptote: y axis

Well this about sums up the lesson that we had today! I hope this helps, even though I know that no one ever looks on this website! For more accurate and beneficial help go see Mr. Cope and take up his free time! (I'm sure he will be very happy about that! :)

Sunday, May 15, 2011

Fitting Exponential Models to Data

On Thursday we learned about fitting exponential models to dada. This lesson was based a lot on graphing and lists and spread sheets, so the Ti-Inspire was crucial. We first looked at how to solve using regression without a calculator.
For example: Using the points (0,1000) and (5, 550)
you use 1,000 as the a value (because its the y intercept) and plug the rest of the equation into
a(b)^x form.
550/1000= 1000/1000x b^5 (multiply both sides by 1/5)
and b=.887.

We then moved on to using the calculator using lists and spreadsheets
1. Add lists and spreadsheets page
-Enter your data given and label the columns
2. Press menu 4,1 (Statistics, stat calculations) then A (exponential regression)
3. Choose your x and y values
4. There should be a table of values including A, B and the function has been stored as f(1) for later use.

To graph and plot the points:
1. Same as above, add a lists and spreadsheets page and enter your data
2. Then press Control i to add a Data and Statistics page
-Here we choose our x and y value to make the scatterplot
3. Click menu 4, 6 (analyze and regression) and then 8 (exponential)
-The scatter plot will appear along with the equation

We have a quiz tomorrow on 9.1-9.4 (Monday May 16th)
What we've also covered:

-Exponential Growth/Decay: y=ab^x
Domain: All reals
Range: y:y>0
Growth:b>1 (when plugging in growth factor remember to add a 1 to the beginning of the percent and turn into decimal...ex: 6.6%=1.066
Decay: 01 (when plugging in grown factor remember to subtract the percent from 100...ex: 88%=.12)

-Continuous Growth/Decay
General function: N(t)=Ce^rt
Continuously compounded: (ONLY if it says continuous in the problem): A=Pe^rt

-Compound Interest
*if it is a growth its a positive number for r, if its decay, r is negative

-Half Lives
Example problem: The half life of Carbon-14 is 5,730 years. If an organism had 8 grams of Carbon-14 and has been dead for 28,650 years, how many grams of Carbon-14 are left?
explanation: You have to divide the 28,650 by the 5,730 to get the half life. The answer is 5 so you think about the original 8 grams that you have to cut in half 5 times. If you divide 8 by 2 five times, you end up with 1/4.

The homework is the Review assignment #1 for 9.1-9.4 and quiz monday!