 In this video, I want to talk about some elementary properties of logarithms, properties that follow immediately from the fact that if f of x is an exponential function, then f inverse is going to be the logarithm base A, right? So, exponential functions are inverses to logarithms and logarithms are inverses of exponentials. And some immediate consequences of those statements would be the following. It is a fact that the log base A of 1 is always, always, always 0. It doesn't matter what the base is, so long as the base is positive and not equal to 1, that gives you an acceptable logarithm, and the log of 1 is always equal to 0. How do we see this? Well, we could convert this into the exponential form. If we move the A over to the right, the base A, you're going to get 1 equals A to the 0. And as we know from exponential rules, if you raise a positive number to 0, you always get back 1. And as such, since 1 is equal to A to the 0, this tells us that log base A of 1 is equal to 0. So, this is an important observation here that for a logarithm, assuming no transformations are in play, x equals 1 is going to be the x-intercept of logarithms. It's likewise true that if you take log base A of A, that gives you a 1. And again, you can translate this over into exponential form and see this fact. If you have any question about the logarithm, switch the logarithmic form to the exponential form. Notice that if you move the A over to the right, you're going to get A is equal to A to the first. Alright, if you raise something to the first power, you get the number back. So log base A of A is equal to 1. Okay, look at this one right here. If you take the log base A of A to the x, this is equal to x. Well, move that base A to the other side and what do you see here? If you move the A to the side, you're going to be left with an A to the x on the left. But on the right, you're going to get an A to the x, which is like, that's true. A to the x, of course, is equal to A to the x. And so if you take this clearly true statement and switch it to logarithmic form, you get this maybe not as obvious statement for those who are trying to learn how to speak logarithmic. Log base A of A to the x is equal to x. This is a very, very powerful statement because when you look at this, you're saying the following. If I take log base A of A to the x, you give me back an x, this is just going to give you the exponent right here. What you're trying to ask yourself is the following. What power? What power of A gives me this number right here? Well, if the number itself is a power of A, you're like, what power of A gives me A to the x power? Well, it's just the x power. And I want you to convince yourself that this statement right here is none other than just the inverse function property that we've talked about previously in this lecture series. What is the inverse function property? Again, remember, if you have a function and you compose it with its inverse function, right, you have f inverse of f of x, this always turns out to be x. So a function composed with its inverse function always gives you back x. And that's what you're seeing right here. If you perform an exponential and then you perform a logarithm, it's as if nothing happened to the number. It just went on a huge, huge circle. But the inverse function property also goes the other way around. What if you take f of f inverse of x? That's supposed to equal x as well. So what if I do a logarithm and then an exponential? Aha, that's what we have right here. What if you take the log base A of x and you raise that base A, you exponentiate it base A, that's going to be the same thing as x. And I want you to think about this for a second. The logarithm is the power, right? That is, when you look at log base A of x, what you're asking yourself is the following. So what power of A gives me x? Okay. So the log base A of x is the power of A that will produce the number x. Well, if I raise A to the power that gives you x, then of course that's going to give you x. That's what the inverse function property is saying right here. You can also, again, if you want another statement here, this right here is an exponential form. If you move the base A to the other side, then the left hand side will just look like the log base A of x. So you get log base A of x. And then as you move the base A to the other side, exponential base A will switch over to logarithm base A and so you get log base A of x equals log base A of x. So again, that's a statement that's trivially true. And so we could see that these properties are true because when we switch it to the other language, the statement has already been established to be true. Now, one thing you have to caution yourself about is this statement right here. Sometimes we get things a little bit mixed around. The log of 1 is always equal to 0, but the log of 0 is actually undefined. And the reason for that is if this number were equal to something, let's say it equaled the number x, right, then you could switch this to an exponential statement and get that A to the x is equal to 0, but this actually can't happen. Exponentials never can equal 0. That's where their horizontal acetote is. And so since 0 is outside the range of an exponential, that means that 0 is outside the domain of a logarithm. We'll talk some more about that in a little bit, but I want to actually show you how you can use property three right here to compute logarithms without the use of a calculator whatsoever. If you take the argument of your logarithm, if you can write that as a power of the base, then it turns out you can use that to compute the logarithm. So for example, okay, if you take the log base 10 of 1000, could we write log base or could we like write 1000 as a power of 10? And the answer is yes, powers of 10 are very easy. If your first digit is a one and you count the number of zeros that gives you the power of 10. So this is going to be log base 10 of 10 cubed. And so what power of so we're asking ourselves what power of 10 gives you 1000. Well now we're asking ourselves what power of 10 gives you 10 to the third power. Oh, the answers in the question, the log base 10 of 1000 will equal three. Okay. Here's another example log base two of 32. What power of two gives you 32 some of us might already know the answer, but let's just rewrite it here 32 is equal to two to the fifth right 248 1632 it's the fifth power there. And so what power of two gives you two to the fifth that's equal to the fifth power, like so. Let's do another example let's take log base 10 of 0.1. What is what what power of 10 gives you 0.1. Now this one we actually can do it might not be obvious as a decimal but if we switch it over to a fraction becomes a lot easier. If we take the log base 10 of one tenth right I mean point one is a 10th. So we get the fraction one over 10, for which when it's in the fraction form, we actually want to write this as an exponential statement. So one over 10, we can write as 10 to the negative one. And so then what power of 10 gives you 10 to the negative one power that's going to equal negative one. Now we have two more questions I want to compute but let me actually make a mention about some notation before we go on. When it comes to base 10 we actually talk about base 10 so frequently, like when one talks about the Richter scale or decibels, or pH factor, you know, a lot of a lot of things on logarithmic scales we use base 10, just because our number system is a decimal number system it's base 10. And so this is when you work base 10 it's often called the common log. And so you might see, you might see me or in other mathematical settings someone just write down LOG and they didn't write down the base. You can assume, you can safely assume that if the base is not written down you see LOG of X that means base 10 of X and this is called the common log. And most scientific calculators will be equipped with some type of LOG log button. That button is used to evaluate the common logarithm. Another logarithm that's very important is what we call the natural log. So the natural log is denoted ln of X. And this represents the log base E where E is that that it's that irrational number approximately 2.7. It's a very important number when it comes to logarithm. So important that when you work base E it's called the natural logarithm denoted ln. And most scientific or graphing calculators will have an ln function or an ln button somewhere on the keyboard. Now, in case you're afraid you're being dyslexic right now, it is intentional that the natural log is denoted ln, not nl. And that the reason is the notation for the natural log actually comes from French for logarithm natural, in which case the words are transposed through order. Let's go back to this. So these ones right here I could have actually denoted this as just LOG of 1000 or LOG of 0.1. Those would have been acceptable because by not mentioning the base we will assume that is we're talking about the common log. Let's do two more calculations here. What if we want to do LOG base 16 of 4? This one can be kind of a tricky one. We're trying to think, what if we're like, I don't know the answer. Let's just call it x for a moment. If it helps you could switch this over to the exponential form and we get 4 is equal to make sure you get the right order 16 to the x right. Some people often confuse this with like, we think of something like this, which this isn't quite right, because the numbers are the wrong spot, the x is the wrong spot. We're looking for what power of 16 gives you 4. Now the temptation is the following. It's like, well, I know that 4 squared is equal to 16. I also know that 4 squared is a great game when I was in elementary school, but 4 squared is equal to 16 here. Now, so the temptation might be like, oh, the answer is 2, but I'm not looking for what power of 4 gives me 16. I'm looking for what power of 16 gives you 4. The order of operations matters here, right? If you take 2 cubed, that's an 8, but on the other hand, 3 squared is equal to 9, right? If you switch the exponent in the base, you get a different result. So we have to make sure not to make that mistake right here as well. But this observation, this observation that 4 squared equals 16 is helpful here, because if you move the exponent to the other side, you get 4 equals 16 to the one-half there. So this is the square root of 16 equals 4, and that's therefore the exponent that we're looking for right here, right? What power of 16 gives you 4, and the answer is going to be the one-half exponent. So we get one-half, or if you prefer a decimal, you get 0.5. And so it's very possible that when you're computing a logarithm that the calculation will involve a fraction or decimal of some kind. So with that in mind, what do we see about this one right here? The log base square root of 2 is equal to 4. Now when it comes to a logarithm, the only stipulation we have on the base is that the base has to be a positive number not equal to 1, for which the square root of 2 is satisfactory there, right? So if we call this x for a moment, what power of the square root of 2 will give me 4? Well, let's try to see the relationship, right? You probably immediately remember that 2 squared is equal to 4. And so how do you get the square root of 2? Well, the square root of 2 squared is equal to 2. So combining these statements, we actually get that the square root of 2 to the fourth power is equal to 4. Because if you square the square root of 2, you get 2. And if you square that, then you get 4. And so the double square, right? The square root of 2 squared, that's where we get the square root of 2 to the fourth power there. And so then the answer to this one is the log base square root of 2 of 4 would equal 4 itself. And so oftentimes we're able to compute logarithms without any calculator whatsoever. We can do this if the logarithm turned out to be a rational number, a whole number of our fraction. Oftentimes though, you do get stuck with something like you have to do like say the log base 2 of 5, right? That's going to be a rational number. And so at some point we probably need to use a calculator. But a lot of these ones we can do without a calculator. And it's a good exercise to try to do them without a calculator to actually help you understand what a logarithm actually is. Because many students kind of run around with basically their eyes on their hands on their face here and they can't see through their eyes. They're like, I don't know what a logarithm is. And so they just run through the logarithmic forest lost, right? If you want to be successful with exponentials and logarithms, you have to understand what a logarithm is actually computing. Now at some point, yes, you're going to use a calculator because the calculation is too difficult. But you still have to know what your calculator is trying to find if we want to be successful computing logarithms. And logarithms are just computing the exponent, computing the power. What power of my base will give me this number x? That's what the exponent is. That's what the logarithm is. The exponent is the logarithm and the logarithm is the exponent. The two are one and the same thing.