 Hi, this is a video discussing a brief algebra view of exponential and logarithmic functions. First, exponential functions. These are functions of the form y equals b times a to the x. The key component that makes something exponential is that the variable x is an exponent. b has to be greater than 0 and a cannot equal 0. So what defines exponential function is your variables and the exponent. When the base a is greater than 1, that means we have an increasing exponential function or exponential growth, as it's called. When a is between 0 and 1, well, that means we have exponential decay, which is a purely decreasing function. Now the log function is of the form y equals log base a of x. It is the inverse of the exponential function. That means if you take the exponential function and you switch the x and y values, they switch places, you'll get the log function. Now, there's an easy way to convert from log to exponential form. So if you have y equals log base a of x, notice that whenever I pronounce my log function, a is called the base. So think about it this way. It's all about the base. a is your base. In exponential form, a will be needing an exponent. So if a is hanging out with x in log form, that means a will be hanging out with the other component y in exponential form. So at the end of the day, you get a to the y equals and now x is all by itself. So a to the y equals x. That's how you convert from log to exponential form. Remember, it's all about the base. Now, just as a little fyi, we cannot take the log of numbers less than or equal to zero. They have to be quantities strictly positive. So let's get some practice. Convert to exponential form. So remember what I said. It's all about the base. So start off, circle the base a. a is hanging with four in log form. That means a will be hanging with five in exponential form. So that means you get four equals a to the fifth. Or you could say a to the fifth equals four. Either format is correct or okay. Next, natural log of six equals y. Instead of the LOG log, this is the natural log. What's so special about the natural log is that it always has a base of E, which is Euler's constant. E is approximately 2.718. So okay, circle your base E. E is hanging with six in log form. E hangs out with y in exponential form. So you get six equals e to the y. Now, just a little note here that when there is no base indicated, the base value of the LOG log function is always 10. Remember, that's when there's no base indicated. And it'll always be E for the natural log function. So it'll always be E for ln natural log. Now, we can also convert from exponential form to log form. So remember, it's still all about the base. The base is E in exponential form. That means I will be dealing with LOG or ln. Anytime you see E, that's ln. The base is E. What's gonna go inside the log? Will it be B or will it be nine? Well, if E is hanging with B in exponential form, E will be hanging out with nine in log form. So natural log of nine equals B. There's no need to write the E as the base because it's understood that natural log always has a base of E. Therefore, natural log of nine equals B. Next in part B, you have the variable A. Base is A. A to the fourth equals 24. So this means that LOG, the base is A of 24 equals 4. So that is converting from exponential to log form. Now, let's get a little bit of practice solving some basic log and exponential equations. First things first, to solve a LOG equation. LOG equation means that the variable was contained within a LOG. You will convert the equation to exponential form. So I'll put a little note here that says convert to exponential form. So similar to what we did on the previous slide. So identify your base. It's three. The quantity 4x minus 7 is contained within your LOG. Three is gonna go to the other side of the equation and become the base for two. So the base will be three, the exponent will be two. So 4x minus 7, the quantity inside the LOG is by itself, and then you have equal to three squared. 4x minus 7 equals 9, and it's your job from this point forward to solve for x. So nothing too fancy here. Solving a basic linear equation now, where I add seven to both sides, and then divide both sides by four. Finally, I get x equals four. Now, an exponential equation means that the variable is in the exponent. So 2x minus one, the variable x is in the exponent. So what we need to do is isolate this exponential piece, this base and this power by adding one to both sides. So we'll get e to the 2x minus one equals six. Now one strategy that can be used is take the LOG of both sides of the exponential equation. So natural log of e to the 2x minus one equals natural log of six. Now, what happens here is when you're taking natural log of e, this is a fact, the natural log of e is equal to one. So what that means here is natural log of e to the 2x minus one is equal to 2x minus one. It's one of the cool facts of LOGs. You're taking the natural log of e, whatever the power is on e, becomes what the expression simplifies to. So natural log of e will simply become 2x minus one equals natural log of six. Our goal now is to isolate x by adding one to both sides of the equation. You cannot combine the six and the one. You cannot combine. You cannot combine the natural log of six plus one. These are separate terms. There's a log term then there's a plus one term. So the last thing you can do is divide both sides by two. So that means I get x equals natural log of six plus one divided by two. This is an exact answer, whereas an approximate answer, if you use your calculator, around the three decimal places would be 1.396. So you have an exact answer versus an approximate answer. And usually they'll tell you which one they want. Sometimes it's both. There are also some cool properties which log functions possess. For instance, property one tells me that if I'm taking the log of a product, I can split it up as to some of two individual LOGs. The log of a product equals the sum of the LOGs. We call this property the product property. For example, the log base 3 of 7x equals the log base 3 of 7 plus the log base 3 of x. Multiplication becomes addition among separate LOGs. Property two will be often referred to as the quotient property. It tells you what to do when you have division within a LOG. Well, it just means that you turn division into subtraction of individual LOGs. The log of a quotient equals the difference of the LOGs. So if you have log base 9 of y over 4, it's equal to log base 9 of y minus you have division log base 9 of 4. And lastly, we have property 3, perhaps the most interesting property of all. That's saying if you have a power within a LOG, so within a LOG, you have a base m to a power r, you can bring that power out front of the LOG. The log of a power equals the product of the power and the LOG. So if you have log base 4 of x cubed, you're allowed to take that power within on the base in the LOG, on the quantity in the LOG, and you're allowed to bring that power out front. It's pretty cool. So we will now use properties of LOGs to write the following as a single logarithm. So what this means is that we will take many LOGs and turn them into one. So some things we can do. When you're condensing, when you're writing as a single LOG, you want to take care of property 3, the power property first. So we're kind of using the properties backwards here. I start with multiple LOGs. I want to get back to 1. So the first thing you can do is you can take that 4 out front of the LOG and you can put it in the exponent of 3. So you have log base a of 7 plus log base a of 3 to the 4th. And if you evaluate 3 to the 4th, you'll actually get 81. So okay, I took the number out front of the LOG, made it the power. Now you have addition of 2 LOGs, I want to write this as 1 LOG. So what's going to happen to the 7 and 81? When you write them in 1 LOG, you multiply them together. So you have LOG base a of course is equal to 567. And that's writing 2 LOGs as 1. It's a similar story in example 5, except notice now we have 2 numbers out front of each of our natural LOGs. So we will move those 2s to the powers of the quantities within the LOGs. So you get natural LOG of 8 squared minus natural LOG of x squared. That's by property number 3. Property 3. Well, that means you get natural LOG of 64 minus natural LOG of x squared. You now have subtraction among 2 LOGs. You want to know how does this look when we write 64 and x squared in 1 LOG? Well, subtraction is associated with division. Natural LOG, natural LOG 64 over x squared. Now let's actually expand some LOGs. When we expand LOGs, we're taking 1 LOG and turning it into many. So it's the reverse of what we just did. There means you will do property 3 last. So first, in example 6, you will have to use property 2 to take that division within a LOG and break it up into 2 LOGs. So you get LOG base 2 of a minus LOG base 2 of b squared, which means lasting you want to do since we took 1 LOG and wrote it as 2. The very last step would be to use property 3 and bring down the power. We will bring down the power of 2 within our second LOG, bring it out front. So that's how you fully take 1 LOG and turn it into more than 1. Lastly, in example 7, you have multiplication between x and e to the x. We want to write this as 2 LOGs. So I have natural LOG of x. Multiplication becomes addition plus natural LOG of e to the x. Now we talked about a property previously. So this was property 1 we applied first. We talked about a property previously that said if you have a natural LOG with the e contained within the LOG itself, remember we already know we have a base of e, but the quantity inside the LOG is also e. That means the natural LOG and the e cancel out, leaving you with x or whatever the exponent is. That's a final simplified answer. Now if you're curious to know a little bit more about the derivatives of the e to the x function, exponential function and natural LOG of x, well the derivative of e to the x is just that. It's e to the x. Of course there will be a chain rule that we have to worry about shortly, but we won't worry about that right now. And then if you have the natural LOG function, the derivative is 1 over x, or more accurately 1 over what is inside the natural LOG. So just a little bit of practice here to get us warmed up is when you take the derivative of our function f of x here, you have 200x. Well that's something we're used to seeing. The derivative of 200x is 200. The derivative of 3 to e to the x is 3. You guessed it. e to the x. So that's kind of fun. Now part b, your function f of x equals 200 minus 3 natural LOG of x. 200 becomes 0 because it's a constant. Minus 3, what's the derivative of natural LOG of x? It's 1 over the inside. So 1 over x. So that means you'll get f prime of x equals minus 3 over x. Of course it's always going to be that simple because the chain rule is going to come and haunt us. So in an exponential function, we apply the chain rule to the exponent. For a natural LOG function, we apply the chain rule to the inside of the natural LOG. So just to kind of let you see this in perspective and we'll get more practice with this later, but you see 3e to the 5x squared plus 4x followed by the term minus 2x squared. Well the derivative still going to be 3e to the whatever the exponent is. That will never change. But the chain rule, the chain rule applied to the exponent, that's my chain rule, take the derivative of 5x squared plus 4x, you'll get 10x plus 4. That's your chain rule. Lastly, you have your minus 2x squared, which is just a term that we're used to differentiating to get minus 4x. This is actually an acceptable form of the final answer. Please note that that 10x plus 4 that resulted from the chain rule just multiplies with the 3 and the base e. Lastly, in part b, we have f of x equals square root of x. Remember that's really x to the half power plus 3 natural LOG plus 3 natural LOG of 9x minus 2. We will first take the derivative of x to the half power or square root of x. You get 1 half x to the negative 1 half. Then you have plus 3. We have to think what is the derivative of natural LOG of 9x minus 2? Well, first it's 1 over what's inside, but then you got to apply the chain rule to that inside. So what's the derivative of 9x minus 2? It's 9. We will write this answer maybe a little bit neater. I'm fine with the 1 half x to the negative 1 half. But if you want, you can multiply 3 by 9 to get 27 over 9x minus 2. That's an introduction about the derivatives of the exponential and natural LOG function. Exponential with base e and natural LOG. Thanks for watching.