 Review for exam 4 for Math 1050, College of Algebra for students at Southern Utah University. As usual, I'll be your instructor today, Dr. Angel Misseldine. Like I said, this is the review for exam 4. This is the fourth and the last midterm exam for our course. It's not the final exam, which will be comprehensive, and we'll talk about that in another video at another time. Exam 4 will basically cover the content and topics covered in chapter 6 in our lecture series, where chapter 6 covers exponential and logarithmic functions. In terms of the lecture series, in particular, it'll cover lectures 36 through 44. Now, as you mentioned, that 36 is somewhat exceptional because chapter 6 actually goes from lecture 37 all the way to lecture 44. Lecture 36 is actually about partial fraction decomposition. This is a topic that really belongs to chapter 5 about rational functions. But because of the content and the timing of exam 3, any questions about partial fraction decompositions actually got postponed to exam 4. So we'll see one question about partial fractions in this exam. The rest of it will be about exponential and logarithmic functions. Because this exam only covers one chapter, in some regard, it's going to feel a little bit shorter. But generally speaking, exponential and logarithmic functions are a little bit more challenging than others. So it does balance out in that regard. We're going to have 10 questions in the multiple choice section. As this exam is a little bit shorter, each of those multiple choice questions actually will be worth six points instead of the usual five. So they'll carry a little bit higher weight. So put more attention to those, of course. The free response section, on the other hand, will be a little bit shorter. There will be only four questions in the free response section. Most of them are worth 10 points each, although question 12 will be worth eight. We'll talk about more of that in just a second. So let's jump to the multiple choice section and talk about the types of questions we're going to see here. Question number one, you'll be asked to compute a logarithmic expression. This will be one you can do without a calculator, something like log base eight of two. Logarithms were first introduced in our lecture series in lecture 39. And among other things in lecture 39, we talked about what a logarithm is. It's the exponent of some exponential expression. So we're asking what power of my base gives me the number and play here. This can be done without a calculator. And honestly, using a calculator will probably only slow you down on question number one. If you understand what a logarithm is, you can answer question number one. Question number two is going to ask us to solve a fairly simple exponential equation. We'll do some more challenging exponential equations later on in the exam. I'll point them out, of course, when we get there. For question number two, though, the critical observation you're going to want to know is the following. If you have a to say the m power, and this is equal to a to the n power, the only way that two exponential expressions can equal each other with the same base is if the exponents are in fact equal to each other. So m equals n in that situation because exponentials are one-to-one functions. That's a critical observation here. That'll simplify question number two, so you don't need to use logarithms to solve it. Of course, other exponential laws would be helpful here. If you have a to the m and you times that by a to the n, that's the same thing as a to the m plus n. That could be useful on this one. If you have a to the m divided by a to the n, that's the same thing as a to the m minus n, also might be useful or a to the m. If you raise that itself to the nth power, that's a to the mn. These other exponential laws can be very useful, not just on question number two, but for many other questions that you will see here. Solving these simple exponential equations was also introduced in lecture 39. As a reminder, lecture 39, it continued on the idea of exponential functions. We solved exponential equations such as this, but we also mentioned that there's a limitation to the equations you can solve, and that's what led to the idea of a logarithm also in lecture 39. Of course, also in our lecture series, lectures 41 and 42, they talked about solving exponential logarithmic equations. Those ones in 41 and 42 are a lot harder than what you'll see in question number two, but you might want to reference them if you want some more practice solving exponential logarithmic equations. Question number three is going to be a curve fitting question, where you'll be given some type of exponential model, and you're going to be given some point, some data about that, about that model, in which case then we want to find the missing parameters. In this case, we'd have to find the base A and the parameter, the initial value, the parameter C there. So we need to plug these things and solve for them. You don't just want to solve some nonlinear system of equations, we can be much more strategic about that using properties of exponentials and logarithmic functions. Lecture 38, we did several examples of graphing exponential functions and more relevant to this one, fitting exponential functions to certain points, all right? Because curve fitting and graphing are very much the same questions. This is something we did in lecture 38, look to there for some more examples. Question four is a very important one that you're going to see, of course, on this test. It's the idea about using laws of logarithms to expand or condense logarithmic expressions. This one asks us to expand it, so we want to spread out the logarithmic expression as much as we can. Maybe break out the fraction, break out the radical, and expand it to maybe something like this or something like this. I'm not saying these are the correct answers, just as a possibility. We want to use the laws of logarithms. We mentioned the laws of exponents above, but logarithms have their counterparts, right? If you take the log base A of m times n, this is equal to log base A of m plus the log base A of n. So we should be able to do that one. If we have the log base A of m divided by n, right, this becomes the log base A of m minus the log base A of n, like so. And then the last one, that again, this might come into play here. If you'd have the log base A of m to the nth power, this is the same thing as n times log base A of m. So using these laws of logarithm, just these three that would be sufficient, we can expand a logarithmic expression. So you have this condensed one, we expand it out to something longer, maybe like one of these, like I said. Or, of course, a variant of this question might ask you to condense it. You might start off with an expanded logarithmic expression and be asked to condense it into a single logarithm, like so. All right? So this example right here, this question is the main idea we had from lecture 41, which was about logarithmic laws, the three laws we see here now on the screening law, one, two, and three. And we use those to expand and condense logarithmic expressions. Question number five, got a little written over, so let me erase that right there. On number five, we're going to be asked to compute the domain of a logarithmic function. So in this situation, we have f of x equals log base two of some rational expression x minus one times x plus one over x plus two. Recall from lecture 40, that's when we talk about this in greater detail. In lecture 40, we learned that the domain of logarithmic function is such that as long as the operand inside of the logarithm is positive, you're going to be a real number. Otherwise, you might be undefined, like in a vertical asymptote or some imaginary number. We're not going to worry about imaginary numbers for this question here. The base doesn't really matter inside of this consideration. The general principle is the following. If you take the log base a of some function g of x, it doesn't matter how complicated that thing is, this is our function f of x, then the domain of f of x is going to coincide with the inequality g of x is greater than zero. So whatever makes the inside of the logarithm greater than zero, that's going to be the domain of the logarithmic function there. So you would take this inequality right here, excuse me, you'd take this rational function, you would set it greater or equal to zero and solve that inequality. So finding the domain of a logarithmic function essentially boils down to solving inequality. This could be a quadratic inequality, polynomial inequality or rational inequality like we see here, maybe absolute value inequality. All of the previous inequalities we've studied this semester could come up in this question that's disguised as a logarithmic question. It really is solving inequality. That's the type of question in play here. Like I said, lecture 40, you could have some examples of such if you need some more practice there. Question number six is going to be another question about solving some type of equation. Unlike question number two, this is going to be a logarithmic equation. It should be, again, on the easier side of things, using laws of logarithms, using laws of exponents and using the basic properties, again, like I said, of logarithms. You should be able to solve these type of equations here. It's useful to know that logarithms and exponentials are inverse functions. To solve this logarithmic equation, you might have to switch from the logarithmic form to the exponential form. That's a possibility, or maybe switch from exponential form to logarithmic form. Again, a very easy log type question here. We did these primarily in lecture 41. Of course, the harder logarithmic equations came up in 42. But this one's on the easier side. You also might want to go to lecture 39 back to the original definition of a logarithm, in particular, focusing on the logarithmic form versus the exponential form. That can be a critical observation for solving many of these equations. So you definitely want to be proficient at switching from logarithmic form to exponential form. Question number seven is going to be another question about solving exponential equations. Again, exponential a little bit harder than the one we saw on question number two. But kind of like question number six, you might need to switch from the exponential form to the logarithmic form to solve it. You'll notice that unlike question number two, the answer here is going to rely upon a logarithm. You can't solve it without a logarithm. So that's definitely going to be necessary here. And even though there's some decimals and the answer is a logarithm, you actually don't need a calculator to solve this one because the answer is going to be in exact form. So you just want to kind of go through the steps, you know, peel the onion, move things from the left hand side to the right hand side until you're left with just an X when you're done. So a good place to look for some more examples here would be lecture 42. Lecture 42 has some more advanced exponential equations, kind of like the one we see here with number seven. But of course, we might also go back to lecture 39, which had the easier exponential equations like question number two. Also, 41 has some logarithmic equations. And again, studying those ones might be relevant for answering the question like number seven. So again, question seven, we're going to solve an exponential equation. We do need to use logarithms, but it's not going to be something super, super complicated. Question number eight is a question about the change of base formulas. Remember, if you have the logarithm base A of X, this is equal to the log base B where it could be any other base of X divided by the log base B of A, the old base. And again, this could be any base. This could be the natural log if you wanted to, the natural log of X over the natural log of A. It could be the common log of X over the common log of A. Whatever you want, this change of base formula will be very helpful so you can calculate things like this. Again, no calculator is necessary. And honestly, the use of a calculator might actually be of an impediment to you on a question like this one. You're better off just understanding the change of base formula and going forward with that. Change of base, of course, was introduced in lecture 41 with the laws of logarithm. So take a look there if you want to see some more examples. And of course, the corresponding homework assignments as well. Question number nine, let me erase this one for a second. Question number nine is going to be a question about applications of exponential functions, which we did lots of exponential models in chapter six. So for example, when we did lecture 37, we focused on just financial applications. So things like compounded interest or continuously compounded interest. So remember that situation, the amount in the account after a certain amount of time, this is the principal times one plus the rate over the number of compounds raised to the NT. This is our compounded interest formula. That will be very, that will be useful for, of course, answering the current variant of the question you see on the screen here. Of course, you could ask about continuously compounded interest or something like A is equal to P to the RT here, where again, P is the principal, the initial amount, R is the rate of growth, T is time, and then you have this natural exponential E right there. We talked about those in lecture 37. Lecture 43 focused on ideas of exponential growth and decay. So we did some examples about population growth. We did some examples about radioactive decay, perhaps of a radioactive isotope. Those use the exact same formula you see right here, although the names of the symbols might have changed. The idea is you have some rate of growth, you have some time, you have some initial amount, and then you have a current amount. Again, change the symbols, it's the exact same formula there. Now, with the population growth and also the radioactive decay, sometimes we talk about things like, oh, the population doubled after this time unit or triples this time unit, or with radioactive decay, we talk about the half-life. In two years, you lose half of your mass. So a variation of this equation might be something like, let's do half-life, for example, you're going to get half of the population where you have T over H, your half-life, right? You still have your initial amount, A0, and this is going to equal the amount you have after some time. So again, this is a slight variation, and you can do some type of change of base formula to switch it from some ER to something like this, because after all, remember this expression, you could write this as 1 half to the 1 over H power, and that's then to the T power, right? And then if you have E to the RT, E to the RT is the same thing as E to the R to the T. Again, you could use this natural exponential E to the R right there, or you can use something like this, oh, it doubles every two years, or the half-life is 10 years or whatever. So there's some variations to these formulas you can do. You should be familiar, of course, with this type of exponential growth or radioactive decay. And then finally, I should also mention that Lecture 44 introduced two other important exponential models that we should be aware of. So there's Newton's law of cooling that told us that the temperature of a material at time T, this was equal to A, E to the negative kT plus TS, where TS, of course, with the temperature of the surrounding area like the colder refrigerator or the window sill that we put the hot apple pie on, the value A, of course, is the elbow room there. It's the difference between the initial temperature of the object and then the temperature of the surrounding area. So you can plug that in for A. So could you set up a cooling type problem? Another problem, of course, you should be prepared for is the logistic growth. So could we do something like the amount after time T is equal to C over 1 plus A, E to the negative kT, where, again, logistic growth is a little bit more complicated. Remember that this A value was equal to C minus A0, the initial amount divided by A0, like so. So that's how you get this number A. K, of course, is the rate. C was the carrying capacity with the maximum population for the culture. And then that's the equation. So what you're going to see on this question, let me come back up here, what you're going to see on question nine, of course, is really just a set up the equation type thing. I don't actually need to do the number crunching, but we do need to know these basic formulas, the compounded interest that we saw before, the continuously compounded interest, and then also just exponential growth and decay, Newton's law of cooling and the logistic growth as well. You should be familiar with these formulas. Put this on of course your note card if you're going to have a hard time memorizing it so that you can answer properly question number nine. On question number 10, we have the graph of a logarithmic function. You have to come up with the equation of a logarithmic function like so. So things to look for, of course, is the location of the vertical asymptote. The vertical asymptote for a standard logarithm should be the y-axis. So the fact that it's been moved tells you something about the vertical shift of the graph. Also, the direction matters. If your function is pointing to the right of its asymptote, there was no reflection. If your function's pointing to the left of your asymptote, that would indicate there was in fact a reflection that happened. And then also, for your standard logarithm, I should say, you have an x-intercept of course at zero. So if we've moved that, that tells you something about a vertical shift. And so with this information in hand, the location of the vertical asymptote gives you an horizontal shift. The distance between the vertical asymptote and the original x-intercept, it's not an x-intercept anymore, but the original x-intercept, that distance measures the shifting, excuse me, the stretching, the horizontal stretching that's in play here. And then the shift upward would tell you the shift right there. Because the general formula you're looking for is f of x equals some logarithm base, whatever, this one's a natural log. You're going to have some type of x minus h over b plus k. This is the general formula here where h and k are your usual shifting horizontal shift for h, vertical shift for k. And then b has the stretch, the horizontal stretch in play there, right? Which is this distance, of course, if you're on the other side, that would make b to negative there. This picture you see on the screen, of course, is a logarithmic function, but this could, of course, also be a exponential function. We did graphs of exponential functions in lecture 38. Lecture 40, we talked about graphs of logarithmic functions. So be prepared to do either one of those for question number 10. You have an exponential function. The general formula is going to look something like this, a times e to the x minus h plus k. You can still see a vertical shift and a horizontal shift. This time you want to think of it as a vertical stretch going on there. The location of the horizontal acetone would give you the vertical shift there. The difference between the original y-intercept and the horizontal acetone will give you the scale. And then you can also see the shift. Very similar to what we did here. Graphing exponentials versus graphing logarithms are very similar because they're inverse functions. So as long as you switch the roles of x and y, you're going to be just fine. Let's move on now to the free response section. Question number 11, which is worth 10 points, will have you solve a more advanced exponential equation. This is the type of stuff we did in lecture 42. Much more involved than the simpler exponential equations we saw in the multiple choice section. It could be one where you have to solve an quadratic-like exponential equation. That is an exponential equation with quadratic form like the one you see on the screen right now. It could be one where you have to solve an equation with two with different bases. So something like 2x plus 1 is equal to 3 negative x plus 5 or something like that. How do you deal with the different bases? Use some logarithms to help you out there. Lecture 42 gave some good examples on what to do to prepare. Question number 12 is only worth 8 points. This is going to be a graphing question. It's very similar to question 10, but now we're going the other way around. So you might be asked to graph a logarithmic equation or a logarithmic function like we did in lectures 38 and 40. So the things you did to study for question 10 are very similar to what you're going to do for question 12. It's just the difference here is now on 10, you were given the graph, you find the equation. This one, you're given the equation, you have to find the graph. Make sure, of course, you do label the graph. You should include the intercept, an x-intercept or a y-intercept if it has any. You should include asymptotes. So vertical asymptotes, horizontal asymptotes if they have any. And then again, I'm not going to say much more because what we said for number question 10 already applies to this one. Moving on to the last page here, we're now on to question 13, which is worth 10 points as well. This is similar to question number 11 in that you're going to have to solve an equation, but this equation is going to be a logarithmic equation. The properties and laws of logarithms will be very helpful here in condensing the logarithm to help you solve it. We did examples like this in lecture 41 and in 42. So see the type of examples there we saw for the more advanced logarithmic equations. And so finally, we then get to question number 14. You might have forgotten, but if you did, I'll remind you, this test does cover partial fraction decompositions that were covered, of course, in lecture 36. So you should be prepared to set up and solve a partial fraction decomposition. Remember, the idea there is to set up the template. Then with the template in hand, you can then set up a system of linear equations and work through that. Or you can also annihilate some of the values. That's also perfectly fine. Not a big deal. Just the type of stuff we did in lecture 36 for partial fraction decompositions. That then brings us to the end of exam four. So I don't really have much more to say about that. Of course, if you have any questions, do feel free to reach out to me, post your questions on Canvas or on this video, or just contact me directly, whatever. I'm here to help. Just let me know. And best of luck on this exam.