 Welcome back to our lecture series, Math 1050, College Algebra for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. This is probably one of the most important sections in Chapter 6 about logarithms, and we're going to talk about the so-called laws of logarithms. By analog, we had talked about laws of exponents a long, long time ago in a galaxy far, far away. But as we translate from exponential forms to logarithmic forms, we can translate those exponential laws into their corresponding logarithmic laws. There have been a few that we've already seen, but these three right here that you can see on the screen are in fact going to be the most important of all of those logarithmic laws. As we describe these three laws of logarithms, throughout assume that a is a positive number and not equal to one, and then choose the numbers capital N, capital B, which does not necessarily coincide with little a whatsoever. Capital N, capital B are going to be positive numbers, positive real numbers and C could be any real number right here. The first law tells us that if we take the log base A of capital A times capital B, this is the same thing as log of A plus log of B. The base doesn't play a huge role in this conversation right now. If we take the logarithm of a product, this turns into a sum of logarithms. I want to convince you that this logarithm follows from a law of X1s we're already familiar with. Recall, if I take A to the M by N power, excuse me, that's not the one I want, one, two, three, so take A to the M times A to the N. We've learned previously that if you have A to the M times A to the N, this is equal to A to the M plus N. I want you to convince yourself that these statements are actually the same thing. You'll notice that with our exponential equation, we have a multiplication that turns into an addition, right? So this tells us that when we multiply, we add together the exponents. The same thing is happening also with this logarithmic law, that with the logarithm, when we multiply, then we add together the exponents, really add together exponents we talk about. You're adding together logarithms. But remember, logarithms are exponents. When you look at something like this, what power of little A will give you A times B? So you're multiplying and asking, what power? Well, the power is going to be the power that gives you capital A and the power that gives you capital B, you add those together. So when you multiply, add together exponents. That's what that law of logarithms says, the same law we saw before. The second law of logarithms tells us that the log of A divided by B is equal to the log of A minus log of B. Now, if you're all on board on law, the first law here, the second law kind of makes sense, right? Because multiplication coincides to adding together exponents. Well, division, its inverse operation coincides with subtracting exponents, okay? And again, this also corresponds to a law of exponents that we've seen in the past. If you take A to the M divided by A to the N, so if you divide exponentials here, you're going to subtract their exponents. We see the same thing happening here. Division implies subtract the exponents. We see it's the same thing here. Division means subtract the exponents because you're subtracting logarithms and logarithms are the exponents, okay? As you translate from exponential forms to logarithmic forms, these two equations say the same thing. And then the third law of logarithms tells us that if we take the log of A raised to the C power, this C actually comes out in front and becomes my coefficient here that the log of A to the C is equal to C times log of A, that is. And so again, the corresponding law of exponents is the following. I have A to the M and you raise that to the N, that equals A to the MN. So you see with this law of exponents here that when you raise the power by a power, you multiply together the powers here. And so you see the same thing. When you raise it to a power, you multiply together the powers. Well, there should be two powers, right? One of the powers was C. The other power is the logarithm because logarithms are powers. And so these laws of exponents become critical in helping us build to solve and solve logarithmic equations, exponential equations and also compute many logarithmic expressions. So what we wanna do next is simplify the following logarithmic proper expressions using those laws of logarithms we saw on the previous screen. So you'll notice here that we have the log base four of two plus the log base four of 32. Now we can actually do this without any logarithm properties whatsoever, we could identify that, oh, what power of four gives me two, the one half power, that's because it's the square root. And what power of four gives me 32, you could get, you could unravel that and determine it turns out to be five halves. So you could take one half plus five halves and end up with a six halves or three, that's perfectly acceptable. But I wanna show you that we can actually do this with laws of logarithms, right? We saw on the previous slide that if you take the log of little a of a times b, this is equal to the log of capital A plus the log of b right here. And so you see if you have a sum of two logarithms which have the same base, you can condense those two logarithms together using multiplication. And so then what we see in this example here then is the log base four of two plus the log base four of 32, this becomes the log base four of two times 32 which that turns out to be log base four of 64 which 64 notice is the third power of four. So what power of four gives you the third power? I will allow b three because the logarithms are the exponents there. So by condensing the logarithm, it makes it a much easier calculation. You don't have to worry about all the fractions and square roots that come into play here. Look at this next expression, log base two of 80 minus log base two of five. So unlike the previous example, these guys are irrational. There is no rational power of two that gives you five. So that would be very difficult to do without a calculator but the second law of logarithms comes into play here. If I take the log of a divided by b, this is the same thing as the log of a minus the log of b. And so when we have the difference of these two logarithms right here, we can condense them together using that second law for which we get the log base two of 80 divided by five for which five goes into 80, 16 times your log base two of 16 and 16 is a power of two. It's the fourth power, two to the fourth of 16. So we get that this turns out to be four. So you have this irrational number minus an irrational number. Turns out it's a whole number, it's equal to four. And then the third law of excuse me, the third law of logarithms told us, remember if you take the log base a of a to the c, this is equal to c times the log of a. So if you have a coefficient in front, you can bring it inside as a, you can bring it inside as an exponent. And so this expression becomes the log, this is the common log of course, the log base 10, you get eight to the negative one third, which negative one third of course means we're taking the log of, well you can take the cube root of eight raised to the negative one power for which the cube root of eight is two, right? So you get the log of two to the negative one, that just means log to the one half for which if we wanted a decimal approximation we could use a calculator here, but this is gonna be the log of one half. This is also the same thing as negative log of two if you prefer. Those are both the same thing because after all you could bring this negative sign back out as a coefficient if you prefer. Either way, this is something you wanna put into your calculator because we can't go any further without it and we're gonna get negative 0.301 as the calculation here. Now, when it comes to these laws of logarithms it's very important to be able to use them to be able to expand or condense logarithms. It's kind of like a plane in the accordion. We pull them apart, we squish them together, we pull them apart, we squish them together. And we have to be like Daniels on here and practice wax on, wax off This is an algebraic skill that's very useful in future mathematical problems such as solving exponential logarithmic equations. It's also a very valuable tool in calculus. So let's first start with the idea of expansion. So the idea of expanding the logarithms, we wanna spread the logarithm out as much as possible into smaller, smaller, smaller pieces. Okay, as it make the log as simple as possible. So when you look at this first one log base two of x times the square root of x squared plus one as you are multiplying these two things together we can use the first law of logarithm and expand it. So you get log base two of x plus log base two of the square root of x squared plus one. So that's what the first law of logarithms tells us we can expand that. So the first one's pretty good. The log of x, you can't really do better than that. The second one's still a little bit more complicated and it's present form it might not be obvious what to do but if I rewrite it, right? When it comes to algebra, we actually get more and more happy with exponents than other symbols. So if you take the square root of x squared plus two that actually is the same thing as x squared plus, excuse me, the square of x squared plus one is the same thing as x squared plus one to the one half power. The reason this is relevant is because the third law of logarithms comes into play for which we can pull out that coefficient and then it becomes again by the third law we get, well, we still keep the log base two of x we had before but then we get one half the log base two of x squared plus one. And this right here is gonna be the expanded form of the logarithm right here. We can't expand it anymore. The second law doesn't apply the first and third we've done as much as we can. Now you might be a little bit wondering like well, what about log base two of x squared plus one? Could that not become two times the log base two of x plus one, right? You might wonder, does something like that hold, right? Can you pull the two out of the logarithm? And so in this situation, the answer is no, those things are not equal to each other, big, flat, you know, big stinking x right there. The issue here is that when it comes to using the third law of logarithms, so the log base A of A to the C, this equals C times log base A of A right here. So to pull up the exponent C, what you have to do is you, it has to be the exponent of the entire thing. You'll notice that the entire expression x squared plus one was raised to the one half power. In this situation, only the first term is raised to the two so we can't pull it out. Now before you, again, you get too excited with what I'm talking about right here. If I had like x squared plus y squared, this still would not equal two times log base two of x plus y, that also would not work. The thing is the entire expression has to be squared, not just individual terms, even if all of the terms are squared, that's not how the third law works. Let's look at another example here. Let's use the natural log in this one. So we have the natural log of x squared divided by x minus one cubed. Well, you'll notice that since we have this rational function, there's division, right? There's a top and divided by a bottom. The second law of logarithms applies right here for which this would then become the natural log of x squared minus the natural log of x minus one cubed. Like so. Now these exponents, x squared, everything that the whole expression was squared, it comes out. The three right here, whoops, that's not far enough. We can bring out the three and we end up with two times the natural log of x minus three times the natural log of x minus one. And this would then be the expanded form of this logarithm. If we can expand logs, we can also go the other way around. We can use the laws of logarithms to either combine or condense logarithms. So if you have this long string of logarithmic expressions, we can condense it all together into one very small piece. Complicated, mind you, but very one single logarithm. This is a very useful skill when we solve logarithmic equations, but also the expansion's a very useful thing when we get to calculus later on. So if we have the expression three log x plus one half the common log of x plus one, the first thing to do is to bring these exponents inside using the third law of logarithms. So by the third law, you end up with log of x cubed plus the log of x plus one to the one half power. And if you don't wanna write as an exponent, we can switch it into a square root. For example, this would be the log of the square root of x plus one. And so then since we have addition now separating the two logarithms, right? Addition with no coefficients, right? We couldn't use the first law first because of the coefficients. But now that the coefficients are gone, this is a sum of two logs. We can combine them together using the first law of logarithms. And we end up with the log of x cubed times the square root of x plus one. And then as a final example of this, maybe have three terms going on here. Just do it one at a time, right? So we're gonna think about, okay, I have these coefficients I'm gonna bring inside. So we end up with the natural log of s cubed. We're then gonna get the plus the natural log of the square root of t. And then we're gonna minus the natural log of here. We're gonna get t squared plus one raised to the fourth power. So when we have three logarithms here, the laws only apply to two at a time. So just look at the first two. You have a sum of two logarithms. So we can bring them together by the first law this time. This becomes the natural log of s cubed times the square root of t. And then you're subtracting from it the natural log of t squared plus one to the fourth. And then because now the two logs are separated by a subtraction sign, we bring them together using the second law. And this becomes the natural log of in the top, we have three squared times, excuse me, s squared times the square root of t. And then the denominator, we're gonna get t squared plus one to the fourth power. And this is then our condensed logarithm. And one nice rule of thumb is that whoever has a positive coefficient when you started will end up in the numerator. And whoever had a negative coefficient when you started will end up in the denominator when you condensed this things together. In terms of expansion, of course, whoever started in the numerator will have a positive coefficient when you're done expanding. And whoever started in the denominator will have a negative coefficient when you're done expanding these things. Now I should of course caution everyone here that if there are laws of logarithms then there are necessarily crimes of exponents. We can go to logarithmic jail if we break these things. And so these are some issues one has to look out for. These are some common mistakes that students make when they're first learning about logarithms. So we have that the first law of logarithms told us that if we take the log of x times y, this is the same thing as the log of x plus the log of y. And in all of these examples, I'm writing as a common log, but this would be true for any base whatsoever, right? So a product inside of a logarithm turns into a sum outside of logarithm. Now the common mistake is if you have a sum inside of the logarithm that does not become a sum out of the logarithm. So you can't just break it up like that. You can't do that. Same thing with differences, right? We know that if you take the log of x divided by y this will equal the log of x minus the log of y that is in fact justified by the second law. But if you have a subtraction inside that does not become a subtraction outside. You have to have a quotient or difference in, excuse me, you have to have a product or quotient inside the logarithm. That will then turn into a sum or difference outside of it, okay? Another thing is we know that something special happens if there's a quotient inside the logarithm. Well, what if you have a quotient of two logs? Well, sometimes we conflate together that a quotient of logs becomes a log of a quotient, but that's not the case. This one over here, the log of x plus y, this actually is related to the second law of exponents. On the other hand, this one right here, this is actually what we would use the change of base formula, which we'll actually see what that is in the next video. So don't confuse those two together. We know that the third law of exponents tells us if you take the log of x to the n this is equal to n times the log of x. Now the problem is if you take the entire logarithm to the nth power, right? You can't pull the exponent out as a coefficient. And if I have log times log that's not the same thing as two times log. You can pull the exponent away from the argument inside of the logarithm, not from the whole thing itself. So watch out for things like that. So like log of x squared, right? If you have log times log, log times log, this is not the same thing as two times log, which on the other hand is the same thing as the log of x squared. So you wanna be careful, am I squaring the logarithm or am I squaring the argument, the operand of the logarithm, it's a very different thing. And now that I'm thinking about, there's probably one more crime of exponents I should mention here. If you take like the log of like x to the n plus one or something like that, this is not the same thing as n times the log of x plus one or anything like that. Sometimes students try to pull exponents out of the inside. You have to have the entire expression raised to the nth power about individual terms in the sum.