Hi, my name is Rebecca Muller. I have a question to pose for you. How would you solve the following equation? If you have 3 to the x equals 9, which is going to be an exponential equation because of the fact that our variable is an exponent, what would you do to solve it? Well, because these numbers are very familiar to us, you probably can recognize what the answer is. You might note that if I write 3 to the x equals 3 squared, which is 9, then we can now equate exponents in order to say that x equals 2. Now this is going to be true because we have a one to one function when we're dealing with exponential functions. However, what if the equation instead read 10 to the x equals 3? What would you do now? Well these are not numbers that are familiar to us. I do not-- I mean, of course we know what 3 and 10 are, but we don't know what power of 10 the value of 3 would be. So we're going to need another method. And that's going to be using logarithm. So for x greater than 0, b greater than 0, b not equal to 1, log base b of y equals x if and only if b to the x equals y. So what we're getting is a relationship between something called a logarithm and what we know about exponential functions. So we're going to have two different formats. We're going to have a logarithmic form, and we're going to have an exponential form for equations. They're going to give us the same information, just in a different form. For instance, if in the exponential form I start off with something like 10 to the x equals 3, which is what we had posed over here, then I can change that to logarithmic form. So notice my base is 10. The exponent is x. And it equals the value of y, which is 3. So just rearranging and writing it differently, the logarithmic form that gives me the same information is log base 10, which is going to be my base b, of the value y, which is going to be the value 3, equals the exponent, which is x. Now what's key here is to note that the logarithm is equal to the exponent. That's going to be what you need to keep saying to yourself in order to become familiar with logarithms. The logarithm is equal to the exponent. Look back at the exponential equation. The exponent was x. Here our logarithm is equal to x. We could also go from the other format. What if I give you a logarithmic equation? For instance, what if I write down log base 4 of x equals 2? If I change this one into its exponential format, then we would need to simply say, what is the base that I'm working with? And, if you get used to saying it to yourself, it's log base 4 when base is 4. What's the exponent? The logarithm equals the exponent. So the exponent has to be the value 2. And that has to equal the other value, which in this case is x. And so the first thing that you want to become accustomed to when you're dealing with logarithms is transferring from one format to the other, from the logarithmic form into the exponential form and back again, so that you can become familiarized with what a logarithm is really telling you. So now let's look at our subtopics. We're going to begin with evaluating logarithms. Then we'll move on to changing from logarithmic form to exponential form, and vice versa. We've already seen some of that. We're going to look at the relationship to exponential functions. And then graphs of logarithmic functions. Finding domains of logarithmic functions. So let's begin with the evaluation of some logarithms. Remember that what you want to think about is that a logarithm is really an exponent. So if I give you the following expression, log base 3 of 9, and I ask you to evaluate it, what I really want you to think about is, I'm asking you to find the exponent. So if you know that a logarithm equals an exponent, when you see log think, "I want to find the exponent." But I have to put on base 3 to end up with the result of 9. That answer is going to equal 2. And of course, the reason for that is because if we change this into its exponential format, we see that 3 squared equals 9. The base raised to the exponent equals the value that we were given there. Let's try another one. What if we have log base 2 of 1/8? Well, I need to figure out what the exponent is I would put on the base 2 in order to end up with 1 divided by 8. Now we know that 8 is the same as 2 cubed. To put it into the denominator, we know we need to have negative powers. So this result is going to equal negative 3. And the reason for that, in its exponential form, would read 2 to the negative 3 equals 1/8. How about if I ask you to take log base 10, and I give you the decimal format 0.01? Well, to really understand this, let's just change it a little bit. What I'm looking at here is log base 10 of 1/100. And so we're back to a similar problem that we had just at our previous example. What is the exponent I have to put on 10 in order to end up with 1 divided by 100, which is 10 squared? And that's going to equal negative 2. And again the reasoning is that 10 to the negative 2 power equals 1/100. And now that brings up something else that we want to discuss, and that is that the base 10 is the key to our decimal system. And so it's used very often. And because of that, because it's used so commonly, we're going to end up-- whenever you see something written like this, log and then 1/100, and you don't see a base written? It's implied that the base is base 10. It's kind of similar to what you became accustomed to when you wrote down some radicals. For instance, if I write this, it means cube root. But if I want square root, you know, we just write the symbol with the radical. Symbol. We don't put the 2 up here. It's understood. So same idea is going on here. This is going to be called, appropriately, the common log. And it turns out that that's one of the keys that your calculator will have. You'll have a key that just reads "log." It means log base 10. Let's do another example where we're going to evaluate logarithms. What if I'm looking at log base e of e to the fifth power? Now, this is again saying, I want to find the exponent. But I have to put on e in order to end up with e raised to the fifth power. So what's the exponent? It's got to be 5. You know, e is one of those things that arises naturally. And so because of this, we also have an abbreviation for log base e. It's called the natural logarithm, or simply the natural log. And the abbreviation for it is ln. So I can rewrite log base e of e to the fifth by writing ln e to the fifth. It means the same thing. And of course the answer is still going to be 5. In general, if you're asked to find the natural log of e to some power a, this is always going to equal the power. It's the exponent I have to put on e to end up with e raised to that exponent power. And so we want to become familiarized with evaluating logarithms in this manner. It's time for a quick quiz. Why can't log base 2 of negative 5 be evaluated? A. It cannot be evaluated because a positive number raised to a power must be positive. B. It cannot be evaluated because 5 is not a power of 2. C. It can be! The answer is negative 32. Choose A, B or C. Your correct the answer is A. It cannot be evaluated because a positive number raised to a power must be positive. Sorry the answer is A, it cannot be evaluated because a positive number raised to a power must be positive. When we consider log base two of negative 5 remember that negative 5 has to be the result of raising 2 to some power. Another way to consider this is to look at log base 2 of negative 5 equal to some number and I'll just call it x. If we take this logarithmic equation and change it to exponential form it reads 2 to the x power equals negative 5. And so we can see if we have a positive base of 2 we raise it to a power the result must be positive it cannot equal negative 5. Now next we're going to look at some simple exponential and logarithmic equations that can be solved by changing forms. Let's return to the one that we started with. That is, if you recall, we had 10 to the x equals 3. Notice that we're looking for the exponent in this equation. So if we change this to its logarithmic form, we will be solving for the exponent. We can rewrite this as log base 10-- I'll go ahead and write it in this time-- of 3 equals x. And recall that that's the same thing as writing log 3 equals x, because that's the common logarithm. Now this is a key that you're able to find on your calculator. It's going to read exactly that. It's going to be l-o-g. When you use that calculator-- and you might want to go ahead and follow along with me, go ahead and punch it in right now-- what you're going to find is that your x value is approximately equal to 0.477. Now what I've done is I've rounded down. But it's always a good habit to check your work. So let's do that next. I'm going to take this value for x and substitute it into the exponent. So we'll have 10 to the 0.477. And now, using your calculator, you're going to come up with an answer that is approximately equal to 2.999. Why is it that you're not going to get exactly 3 back? Well, remember that right here we ended up rounding down. We ended up dropping off, or truncating. And for that reason, we're not going to actually have the exact value for the exponent. In fact, if you were asked to give an exact value, you would have to say the exact answer is the log 3. This is an approximate answer. Many times when you're working with exponential and logarithmic equations, you may be asked to give both versions, the exact answer and the approximate answer. Here's another example. This is going to be a logarithmic equation to start with. If we have the natural log of x minus 1 equals 2, how can we solve this equation? Well, we can change it into its exponential format. So remember ln stands for natural log, which means log base e. Now, sometimes students get this confused, and they think that ln equals e. There's no such thing. ln is a notation, so it doesn't equal a number. But it does have that relationship. If I change this to its exponential form, what we get is a base e raised to the second power, because a logarithm equals an exponent. And that equals x minus 1. We can solve for x by adding 1 to both sides of the equation. So x is going to equal e squared plus 1. Again, you can use your calculator in order to come up with an approximate value for this. And when you use your calculator, you come up with x approximately equal to 8.39. Again, here's your exact answer. This is your approximate answer. If you want to check this answer, what you're going to do is substitute 8.39 in your original equation. So to check, you're going to have the natural log of 8.39 minus 1. Now, when we came up with the value of 8.39 off of the calculator, it turned out that we were rounding up. So when we substitute natural log of 8.39 minus 1, you're going to come up with an answer that is approximately equal to 2.0001. Because you rounded up, your answer is going to be slightly larger than what your exact answer is, because of the fact that it's an approximation. So these again are called, kind of, simple exponential equations and simple log equations, because they can quickly be changed to the other format in order to find the result. Next we're going to move on to logarithmic functions. And so we're going to start with the graph of logarithmic functions. To do so, I'm going to remind you about exponential functions, because we know there's a relationship. And with exponential functions, you might recall that the definition is that we have f of x equals b to the x. And we have the stipulation that our b value has to be positive, and that it's not equal to 1. If I'm looking for the inverse of this function f of x, then recalling our process we can interchange our x's and y's. So if I start off by writing y equals b to the x, then I can interchange the x's and y's and write x equals b to the y power. Now this is actually going to be the inverse function, but it's not in the format that I want. I'd like to solve for y. Well, where is y in this equation? It's part of the exponent. How can we solve for exponents? That's exactly what logarithms will do. So I'll take this exponential function and change it to its logarithmic form. We're going to have log base b of-- well, we know the logarithm has to equal the exponent. So what that tells us is this is log base b of x. Now what I've found is actually f inverse of x. So if we start off from the beginning, we notice that the exponential function with base b is going to have as its inverse the logarithmic function to the same base. We're going to use that now in order to come up with a logarithmic graph. It's time for another quick quiz. The graph of a function and its inverse are reflections of each other across: A. The x-axis. B. The line y equals 0. C. The line y equals x. Choose from A, B and C. Your correct, the answer is C. Sorry the answer is C. The graph of a function and its inverse are reflections of each other across the line y equals x. So for example, we're going to look at a function that should be familiar to you by now. That is f of x equals 2 to the x. Now for this function, let's just point out a few things. We know that, for instance, that we have points on this graph. And we can say, well, OK, we have the points 0 comma 1. And we can see that we have the point 1 comma 2. We know that there is a horizontal asymptote whose equation is y equals 0. And we can point out that the domain of this function would be negative infinity to infinity. And the range of this function we can see is going to be 0 to infinity. And again, all of this is for our function y equals 2 to the x. Now we just determined that the inverse of this is going to be the logarithmic function to the same base. So, you know, what do we know about this logarithmic function? Well, we know that we can write its equation as y equals log base 2 of x And what's going to have to be true about this particular function? Well, because it's an inverse, what do we know about points on this graph? We know that we can interchange the x's and y's. So we can have the points 1, 0 and 2 comma 1. You know, what's going to happen with the asymptote? Well, again thinking about interchanging x's and y's, we had y equals 0. So now we're going to have x equals 0. Well, x equals 0 is going to be an equation of a vertical line. So instead of a horizontal asymptote, we will have a vertical asymptote. What's going to be true about the domain and the range? Well, because they're inverses of each other, what's going to happen is those are going to switch. In other words, the domain of this logarithmic function is going to be the range of the exponential function to the same base. Which is going to be the 0 to infinity. The range is going to equal all real numbers. Let's look at it graphically and then talk a little bit more about this. You know, the graph of y equals 2 to the x is going to be reflected across the line y equals x to come up with the graph of its inverse. And using what we just had there, we can see that the graph of this logarithmic function, log base 2 of x, is going to end up having that vertical asymptote at x equals 0. It's going to go through the point 1 comma 0 and the point 2 comma 1. And then again-- kind of turn your head a little bit and you can see that this is going to be the reflection across that line. Now one thing that I want to point out before going any further is, if we think about the domain and range, let's make sure it makes sense to us. The domain is going to be only positive values. Going back to this format, notice that this x value is the result of raising 2, the base, to a power. So a positive number raised to a power is always positive. So if I take 2 and raise it to a power, I know that the result has to be positive. That's what this domain is telling us. The logarithm equals the exponent. And we don't have restrictions on exponents. So the range, which is going to equal to the y values, will be the exponent values. And that means we get all real numbers for that. Now, as with other graphs that we've worked with, at this point you should be able to do transformations of a basic function. So the function we're going to be considering next is the graph of f of x equals the natural log of x minus 1. How do you begin graphing that? My suggestion is that you start off by thinking about the domain of that function. And we just saw that when we're dealing with domain, we have to make sure that the value for x is going to make sense in terms of the fact that it's a result after raising a positive number, in this case e, to a power. So this x minus 1 is going to have to be a positive value. So x is going to have to be greater than 1. Now, to figure out what this graph looks like, we know that it's a shift of the graph y equals natural log x. So I'm going to first make sure that we understand what that graph looks like, y equals the natural log of x. And what we can do is quickly sketch that out. Now, one way to do this is to change it into its exponential form, find those values, and then switch them back. Another thing that would be nice is just to get used to trying to use the idea that we had before. That is, what if I have the same base, and what if I have the value 1? In other words, on a little T-chart, how can I end up-- what happens if I take the natural log of 1? Well, let's go ahead and substitute it in and see what that means. We have y equals the natural log of 1. Changing that into its exponential form, we'd have e to the y power equals 1. What would the exponent have to be in order for that to be true? It would have to equal 0. And so you can see that 1 comma 0, as we saw in our previous example, is going to end up on the graph of log base b of x, regardless of the base. So we're going to have 1 comma 0 on that graph. The other value that's nice to have is the same as the base. In other words, what if I put in the base that we're given in the problem? In this case, our base is e. So what I'm looking at is y equals the natural log of e. Changing that to exponential form, we end up with e to the y equals e. What is going to be the value of y to make this true? It's going to have to equal 1. Now e, recall, is about 2.7. So if I come over to my graph, and that's a 2 and that's a 3, I'm going to go around 2.7, and then up 1. And that's going to be my picture. And I'm only going to graph 2 points, but it's going to be enough. I know the domain for this particular function is going to be the values of all x that are greater than 0. And I have a vertical asymptote at x equals 0. So connecting these dots, I'm going to end up heading off this direction, and then coming back here. Now that we've got the graph of y equals natural log x, what is going to occur when we're going to have the natural log of x minus 1? That's just going to be a transformation where we're going to shift to the right 1 unit. That's what that minus 1 is going to do when it affects the x. So I'm going to shift this whole thing right. So I'm going to take this picture and just simply move it. First of all, we had the domain, which was going to be only values of x that were greater than 1. That's a key to the fact that we're going to have that vertical asymptote at x equals 1. So right here. And notice that makes sense. It's a shift of the vertical asymptote at x equals 0 1 unit to the right. What's going to happen to the point that used to be right here at 1, 0? It's going to move right 1 unit, and now it's going to be at 2 comma 0. What's going to happen to the point that used to be at e comma 1? It's going to move 1 unit to the right. And so if it was 2.7 comma 1, 1, 2, 2.7 comma 1, it's now going to be approximately 3.7 comma 1. And we just-- we're going to connect the dots and draw in our graph. And now this is going to be the result of taking the natural log graph and shifting it 1 unit to the right. So paying attention to the domains of logarithmic functions is not just important for graphing. It turns out that when we come up with logarithmic equations, paying attention to the restrictions on logarithms will be important there, too. Let's now look at the domains of a couple of logarithmic functions. And again, that will transfer over to another section that you're working on in the future. So the first example is, if I look at the function g of x equals log base 4 of 2x plus 3, and I'm interested in the domain of this, well, remember that we're going to have the 2x plus 3 as the result of raising 4 to a power. And so we know that when we take a positive number and raise it to a power, it must be positive, which means that 2x plus 3 has to be greater than 0. When we solve this inequality, we will have the domain of the function. So we subtract 3 to get 2x is greater than negative 3. We divide by 2 to get x is greater than negative 3/2. And we can write that in interval notation as parentheses negative 3/2 comma infinity. And that would be the domain of the function g of x. As a second example let's look at h of x, which equals the natural log of 5 minus x. What would you do if you were asked to find the domain? Were going to mimic what I did before. Remembering that this expression, 5 minus x, is the result that we get from raising the base e to a power. So the 5 minus x has to be greater than 0. I can add x to both sides of this equation and come up with 5 is greater than x. Which, if you like to have the x on the left instead, just says x is less than 5. So the domain for this h of x is going to be from negative infinity all the way to the value of 5, not including 5. So again, finding the domains of logarithmic functions will serve you well in this section, but also in sections to come. It's time now for you to try some more problems dealing with logarithms and logarithmic functions. Good luck.