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!!