 Let A be a positive number other than the number one itself. Then we define the logarithm base A, which is denoted LOG, so we often call them logs for short, LOG sub A of X. So the log base A of X is the inverse function to the exponential function f of X equals A. And so these two functions are inverses of each other. So let me kind of, before we talk more about logarithms here, let me talk about other inverse relationships we're familiar with. So let's say for example, we had the expression, you know, X plus three equals five. We'll make it an equation here. If we wanted to solve the equation for X, what we think of it's like, hmm, I gotta get rid of this plus three, but that plus three is attached to the X in some regard. And so the idea is I wanna move it to the other side so that X is isolated on the left hand side. But how do you move plus three? Well, you apply the inverse operation to both sides. So you subtract three, subtract three. The inverse function has the property that addition by three followed by subtraction by three will give you nothing that is just addition by zero won't do anything. And so then the left hand side will become X. And the right hand side would look like five minus three, which we can then simplify to make that computation. So addition and subtraction are inverse operations of each other. Well, what if we did something like three X is equal to five. Now on the left hand side, the goal is still the same. We wanna move the three to the other side of the equation. We wanna get X all by itself. But how do you move the three in this situation? We're still getting rid of the number three, but the three is attached to the variable X by a different operation this time. The operation in play is multiplication. So we will perform the inverse operation so that multiplication by three cancels out with division by three and that then gives us the solution X equals five thirds like there. You can simplify the fraction if it were possible to the five thirds. That's already in lowest terms there. Let's try this one more time here. This time let's consider the equation we're gonna take X cubed is equal to five. Or if you might like to, you think of it as like X care at three, that's oftentimes how you write it on a calculator. There's some like a little carrot or something here. That's the operation in play, the exponent. Well, you have X cubed is equal to five. The same idea is true for all three of these examples so far. If you wanna solve for X, we gotta move, we gotta move the three to the other side, but we can't just magically make it disappear and it reappears in this side. We have to perform some operation to both sides of the equation so that equality is preserved. We have to apply the inverse operation. So if you're taking the power of three, the inverse operation as we've learned before would be the cube root for which we take the cube root of both sides and see that the solution ought to be X is equal to the cube root of five like so. And so that's how we see it in all of these different examples. If you have X plus three, you're gonna subtract three from both sides and end up with the solution of two, five minus three. If you had three times X, you're gonna divide both sides by three and get the solution five thirds. If you had X cubed, then you'll take the cube root of both sides and get the solution as X equals three fifths. All right, now we're gonna get to the heart of the matter, okay? If you had something like the following, if you had three to the X is equal to five, we still want to solve for X on the left-hand side. We gotta get rid of the three. That is, we have to move the three to the other side of the equation and how do we accomplish that? How do we move the three to the other side? Well, we have to perform the inverse operation, but this time, the operation is not the power function X cubed. The operation is the exponential function base three. And so to move three to the side, we have to apply the inverse operation, which in this context, the inverse operation would be the log base three of both sides. And so upon doing that, the log base three cancels with the exponential base three and then you end up with X is equal to the log base three of five. And so that's what logarithms are. They're the inverse operations to exponentials. If you wanna move a plus three, you subtract three from both sides. If you wanna move a times three, you divide both sides by three. If you wanna move a power three, you take the radical three each time. If you wanna move the exponential base three, you'll take the logarithm base three. And that's the pattern that we see over and over and over again when you work with these equations. We have to move the number to the other side to solve for X. And we do that by applying the inverse operation. Therefore, if we have an expression like a function like f of X equals two to the X, its inverse function would be the natural log or excuse me, the log base two of X. The natural log is something we'll talk about later. And so if you had like two of X two to the X equals five, then the solution to that equation would be log base two of five. But conversely, if your function itself is a logarithm, let's say G of X equals the log base seven of X, then its inverse function will be seven to the X. And so if you had like log base seven of X is equal to two, right? Then you wanna move the base seven to the other side and you would get that the solution to the equation will be seven squared. We'll talk some more about that here on this slide, which when you're working with a logarithmic or an exponential equation, we often refer to these logarithmic forms and exponential forms. That as you look at these two equations right here, these two equations are equivalent. They have the same solution set. But this version right here is the so-called logarithmic form of the equation. And this equation right here is the same as what we call the exponential form. And this is something we do all the time. Like we said earlier, X plus three equals five. This is the addition form, X equals five minus three. That's the subtraction form of that equation. Another example, we had three X equals five. This is the multiplication form of the equation. And then you can move multiplication to the other side by division. You get the division form of the equation. All right, one more. If you have X cubed equals five, this is the power version of the equation. Or you have X equals the cube root of five. This is the radical version of the equation. We can do the same thing with logarithms and exponential forms. It's important to be able to convert back and forth between them, as if we were translating one language, like from Spanish to French or something like that. So imagine we have the following exponential statement. Let's say that we have two cubed is equal to eight. So this is a exponential expression and maybe we wanna move base two to the other side of the equation. If we did that, the corresponding exponential, excuse me, the corresponding logarithmic form would look like three is equal to the log base two of eight. So you'll notice that it's the base two that moves to the other side. The base two started on the left-hand side, then it moved to the right-hand side. On the other hand, the three didn't move, it stayed on the left-hand side. And the eight doesn't move, it stays on the right-hand side. It's the base two that seemed to move from the left over here to the right. Let's do another example, but let's start with the logarithmic form this time. If we take log base two of one eighth, this is equal to negative three. Why is that? Well, if you're not convinced that statement is true, we could consider what happens if we move the base two to the other side. If we move the base two to the other side, you're gonna get one eighth is equal to two to the negative three. And remember by exponent laws, a negative exponent means to take reciprocals and two cubed is eight. So this exponential expression, this exponential form is probably more like our native language, all right? This makes sense to us, this is how exponents work. Well, this statement in exponential form is true only if this logarithmic statement is likewise true. So the log base two of one eighth is equal to negative three because two to the negative three is equal to one eighth. Let's look at one more example of this. Let's take the log, the log base five of 25, that is equal to two. Why is that? Well, again, if you wanna think of like moving the base five to the other side, if you move the base five, you're gonna end up with 25 is equal to five squared, which is again a statement, this exponential statement is one we're all gonna probably agree with, right? Five times five is equal to 25. Now because five squared is equal to 25, this means the log base five of 25 equals two. Essentially when you're working with logarithms, you're trying to answer the following question. If I have my base and I have my operand right here, the argument of the logarithm, log base five of 25, you kind of erase this number, right? That's kind of like erasing this number right here. And so you're asking yourself, what power? What power of five gives you 25? That's what the logarithm is trying to compute. What power of five gives you 25? Well, surprise, it's two, right? What power of two gives you one eighth? It's negative three. What power of two gives you eight? It's three. The logarithm is the power. This right here is the logarithm. The logarithm is the power. That's what it's trying to compute. The logarithm is the power.