Weekly Review

During class on Monday we did an online practice with the Properties of Logarithms. An important thing to remember when doing these problems is that logarithmic functions and exponential fuctions are inverses of each other. There were also very important properties to follow when solving these types of problems. They've been listed below:

- Product Property of Logarithms:
MN=B^(x) x B^(y)
MN=B^(x+y)
logB(MN)=x+y
logB(MN)=logB(M) + logB(N)

- Quotient Property of Logarithms:
M/N=B^(x)/B^(y)
M/N=B^(x-y)
logB(M/N)=x-y
logB(M/N)=logB(M)-logB(N)

- Power Property of Logarithms
M^(k)=(B^x)^(k)
M^(k)=B^(xk)
logB(M)^(k)=kx
logB(m)^(k)=k x logB(M)


In simpler words, when expanding a logarithmic expression you want to change any terms being multiplied to addition. For example, log3(11x) expanded becomes: log3(11) + log3(x). It's exactly the same if it's subtraction, except you change it to division. You do just the opposite when condensing a problem. Since you want to make this expression smaller, you're going to switch from addition or subtraction to division or multiplication. For example, log10(4) - log10(x) would be condensed to log10(4/x). Pretty simple.


On Tuesday, we looked further into Solving Exponential Equations. There were 3 different types. Type A requires you to make the bases the same. These are the easiest to do. For example, 4^(x)=8. You would make the bases in each 2 and the problem would then look like this: (2)^2x=(2)^3. From there you make the exponents equal to each other and the final answer is, x=3/2. Type B requires you to take the invers of both sides (log or ln). An example of this type of problem is, 3^(x)=5. The next step would read: xlog3=log5. As you can see, I've moved the exponent (x) and made it the coefficient. Then I took the log of each side. From there, all you have to do is divide each side by log3 in order to get X alone. The answer comes out to be, x=1.46. The third type of problem is a bit tricky, because you don't know both bases. Here's an example: 4x^(2/3) - 5=20. First, you want to isolate the 4x^(2/3), so you add 5 to both sides. Once you get 4x^(2/3)=25, you divide by 4 to get X alone. Since you get x^(2/3) you have to raise it to the reciprocal power (3/2). You raise the other side of the equation to this power. The result is: 15.625 or 125/8.

Also on Tuesday, we looked at how to Solve Logarithmic Equations. Again, there were 3 different types. Type A provides you with a single same base log on each side. For example, log5(3n)=log5(n+2). For this one, the logs cancel and your're left with: 3n=n+2. Next, you just subtract the n from both sides and you get 2n=2. The final answer is, n=1. Type B provides you with a single same base log on one side. An example is, log2(5x-7)=3. You'll need to change this to exponential form. Which makes it turn out to be 2^(3)=5x-7. From there it's easy and you just solve algaebraically. The answer will be, x=3. The third and final type provides you with one side of the equation having more than one same base log. For example, log6(x+1) + log6(x)=1. This would than turn into log6(x+1(x))=1. Just like condensing. Next you eliminate the log6 and what remains is: 6^(1)=x^(2)+x. From here, the 6 is moved to the other side and the equation is set to zero. You factor it to (x+3)(x-2). The answer is, x=-3,2.

No class today and I won't be in for the rest of the week. That's all for my blog!!

Personal Growth

Coming into this class I knew I would struggle. I was never strong when it came to Algebra. I totally forgot everything I learned before and it was basically like starting from scratch. I just arranged with a Villanova College student to tutor me. I'm hoping that will result in me doing better in this class. Not only that but I need to refrain from losing focus during class. That's causing me to miss out on some things. The tests and quizzes are really important to do well on. That's what I'm aiming for. Tomorrow is my first day getting tutored. Hopefully Tuesday's quiz will be the start of me doing better. I also plan to try to stop procrastinating. It's best to study a few days in advance for a test so you'll know what you need more practice on. I've been waiting until the night before and than doing badly on the tests. Once I do all these things I should be doing better in this class.

Semester Review

One of the subjects we focused on this first semester is slope. What is slope? Slope is the ratio of the vertical changes to horizontal changes of a line. It indicates steepness of a line as well as direction. How do you find slope? To find slope you pick any two points on a line (x,y) and (x,y). You then use the rise/run equation (y2-y1)/(x2-x1).

Real Life Example:

Building roller coasters has a lot to do with slope. Roller coasters rely solely on the force generated by the slope of the first hill to glide it to the end. The best slope is one with a curve at the bottom with a flat path. The second slope is known as the “exit path”. The exit path is the slope that helps the roller coaster maintain its velocity after traveling down the first hill. The exit path should have a low slope for a safe exit out of the first hill. This pattern continues for how ever many drops there are.

Late!!!!!! ahhhh.....

In the real world a lot of things relate to what we're doing in class. Besides bridges and tunnels many buildings are built based on parabolas. I named a few in my previous post. When I graduate high school I want to major in music and minor in psychology. I don't really see how math connects to either one. But I guess you could say music and math somewhat relate because with music you have to have rhythm which is involved with timing (counting, measures etc.). Outside of class I like to shop, write poetry/music, play instruments, go to parties and hang with friends. What we're learning in class somewhat applies to my music life as I just said. Also with when I'm shopping and have to deal with sale prices and stuff like that. Other than that I can't see a correlation. That's all for this one!

Parabolas Everyday Part I

Parabolas in Everyday Life:
1. Stain glass windows in churches
2. The Sydney Opera House in Aussie
3. St. Louis Gateway Arch

Stain glass windows were commonly used to symbolize religious happenings in churches and cathedrals. The windows were decorative and informative and many times were donated by members of the congregation in memory of loved ones. These windows also help to monitor how much light is able to shine into the building. Not all stain glass windows are parabolic but those that are need more attention when being built. One incorrect measurement could cause the glass not to fit properly, or to fit but not sturdily and cause injury. I think this shape is used for more of a decorative aspect and is just more elaborate than the usual rectangular shaped window.

The Sydney Opera House was designed by Jorn Utzon and opened by Queen Elizabeth II on October, 20 1973. It conducts 3,000 events each year and has an annual audience of 2 million for its performances. It’s only been open since 1973 but is just as representative of Australia as the pyramids are to Egypt. This building wasn’t purposely built with parabolas in place. Utzon had entered an anonymous competition for an opera house to be built in Australia. Out of about 230 entries his concept was selected. He didn’t even complete the project himself because he left it in the hand of Peter Hall. He along with others completed the interior of the Opera house, but it was all coincidence that it’s shape contained parabolas.

The St. Louis Gateway Arch is the tallest national monument in the US (630 ft). It was completed on October 28, 1965. Each year, about a million visitors ride the trams to the top of the Arch. They’ve been in operation for over 30 years and carried over 25 million passengers. The arch is a catenary curve, meaning if a parabola were rolled along a straight line it would trace out a catenary. The word catenary derives from the latin word for chain. A flexible curve, supported at both ends, that hangs and is acted on by gravity is a catenary. This shape allows for the St. Louis Gateway Arch to be able to move up to 18 inches (wind at 150 mph). It is built to withstand any type of destructive storm.

Links to Pictures:

http://staytondailyphoto.com/photos/stained_glass_church2.jpg

http://www.underthesoutherncross.org/assets/images/australia-sydney-opera-house.jpg

http://www.orgsites.com/mo/stlchristianhomesports/StLouisArch.jpg

End of Quarter Self Reflection

• Assignments-I did all of the work required
• Classwork-I did most of the work required
• Homework-I did most of the work required
• Studying for quizzes-I didn't do much
• Studying for tests-I did most of the work required
• Reworking problems that you misunderstood-I did most of the work required
• Making up work & checking the eboard when absent-I did all of the work required

1. I've done a good job of keeping track of what I need help in. A lot of times I come in for extra help after school to stay updated. Or, if I don't come in for help I ask a friend to explain something to me. Also, the work that I was missing, I made sure to get it completed and questions I didn't understand were explained to me.
2. I could defnitely pay more attention during class to what is being explained. I could also make sure to at least do some of my homework instead of not doing any of it. My biggest problem is not studying enough for tests/quizzes. I come in for help but after that I don't do much studying and that's where my biggest downfall was this quarter.

Blog 5

I'm struggling the most with stating the domain and range by using a graph. My main problem is writing the greater than or less than signs. I keep switching them for some reason. I also am having trouble with f(x) functions when variables are put in the equations.

So far, I think blogging is a good way for you (Ms. DiTullio) to keep track of all of your students. But sometimes they are easy to forget about and could be unecessary points deducted. For example, right now. I just remember this was due and quickly signed on.

We get plenty of worksheets and practice in class and for homework. The only thing I think would be more affective is taking more notes down in class to have as a referance when we get stuck on problems. Instead of asking for help we can look at our notes first than ask.