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.