- Regarding Inverses, I understood the concept of "one-to-one" fairly well. The partner work really helped me understand that to find the inverse of a function one must simply switch the output with the input and it must pass the "horizontal line test" to be considered one-to-one. [That being said, I understood all of homework C1.]
- Also regarding Inverses, I understood that when you find an equation for the inverse function it is very helpful to verify your answers. "f( f-(x)) = f-(f(x)) = x" [Verifying makes me feel better than looking at the back of the book to check if my answers are correct.]
- Moving on to Logarithms, I understood the "story" Ms.Hwang showed us on the board. I thought it was very cool and it helped me understand how to get rid of the logarithm and solve for a variable. [However, I still think I need a bit more practice on solving for variables when there are logarithms in the equation]
- I also understood the rules of logarithms such as the product rule, quotient rule and power rule.
First of all I have to say that some of the questions on homework C2 gave me a really hard time.
- One thing I do not really understand are solving equations with "e". For example (Homework C2 pg.44 #35 & 36). This question asks to solve the equation algebraically and to support the solution graphically. The equation is e^x + e^-x = 3. This is how I tried to solve the problem:
lne x + lne -x = ln 3
x + -x = ln 3
loge x + loge -x = loge 3 (loge)
I didn't know how to multiply ^^^^ that so I gave up.
On #36 I solved the equation further but only got up to: log2 (-x) = log2 5/2
- Another thing that I completely did not understand was how to graph a logarithm equation. Because I did not know how, I was not able to answer straight-forward questions such as "finding the domain & range"-- This was very frustrating for me.
This practically covers most of what I did and didn't understand regarding Logarithms and Inverses. P.S: If you know how to solve any of these, please help me!
hehe, #35, that caused a number of problems on my part as well before i actually solved it. ok, here's how i solved it. first, you simplify e^-x (1/e^x), and after some manipulation, got e^2x-3e^x+1=0. after that, you solve for e^x by completing the square (e^x takes the place of x in this case), and make everything as part of a natural log to get rid of the e (i leave the calculations to you). number 36 can solved in pretty much the same manner, just with different numbers.
ReplyDeletegraphing logarithms isn't hard. there's 2 ways to do it: 1)use a graphing calculator (the simplest way) 2)to graph them on your own, all you really need is 2 points. the first is (1,0) (assuming it is a normal logarithmic function, meaning it has no transformations), because anything to the power of 0 is 1. the second point is (x,2), where x is the value of the base of the logarithm (plug 2 into the y for any ordinary logarithmic function and you see why that is true). after that, you can visually imagine and graph the remainder of the function (it looks just lie the exponential function in terms of shape, just in a different position). if you want to plug in (x,3) into the graph to make it more accurate, that's fine, but to generally sketch a logarithmic function, only the points (1,0) and (x,2) are needed (at least those are the only ones i use, if you need more points, plug in x values and get more until you yourself can make the graph). if there's transformations...yea, i have no trick to those, just take the transformations one step at a time. as for the domain and range, you don't really need the graph to be able to determine the domain and range. if you know the domain and range of the original exponential function, just switch them to produce the domain and range of the logarithmic function (domain becomes range and range becomes domain).
You can graph a logarithm by graphing its inverse in exponential form and then using the y=x line to graph the log as its inverse for exmple graphing the log- f(x)=log base 3(x)you would start out by graphing f(x)=3^x and then graph the line y=x and then just graph its inverse symmetrical to the line y=x
ReplyDeleteLOL no I didn't solve 36 yet! I will later :]
ReplyDeletehmmm i had questions on the same topics!
ReplyDelete