08 - Solving Exponential Equations - Part 1 - Solve for the Exponent - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . The title of this lesson | |
| 00:02 | is solving exponential equations . This is part one . | |
| 00:06 | In fact I really should say this is solving elementary | |
| 00:08 | exponential equations are basic exponential equations because what we're gonna | |
| 00:13 | do in this lesson is introduced a method of solving | |
| 00:16 | these exponential equations . That using the knowledge that we | |
| 00:19 | have up until this point . But a little bit | |
| 00:21 | later when we get to logarithms will have a much | |
| 00:24 | more powerful method of solving exponential equations and when we | |
| 00:28 | get to algorithms will be able to show you how | |
| 00:29 | to do that . But here in this lesson we | |
| 00:31 | can certainly solve a lot of exponential equations using the | |
| 00:34 | knowledge that we have up until this point and by | |
| 00:36 | now you know , exponential functions which we covered in | |
| 00:39 | the last lesson are really and truly one of the | |
| 00:42 | most important class of functions in all of math and | |
| 00:44 | science , in engineering . Everything from nuclear uh nuclear | |
| 00:49 | processes to population dynamics to savings and investments and all | |
| 00:55 | additional applications will be talked about in the last lesson | |
| 00:58 | . So solving exponential equations really applicable to all of | |
| 01:02 | those fields , which is every field essentially . So | |
| 01:05 | the first thing we need to do before we can | |
| 01:06 | show you how to solve these equations , I need | |
| 01:08 | to go over one little property of the exponential function | |
| 01:11 | that's important for you to know . You should know | |
| 01:13 | now that the exponential function in your mind , you | |
| 01:16 | should know what it looks like . It starts small | |
| 01:17 | and it gets very large often when it starts to | |
| 01:21 | kind of go around the corner there and start to | |
| 01:22 | go vertical . It's a 1 to 1 function . | |
| 01:25 | That means for every one input you only get one | |
| 01:28 | output , It passes the horizontal line test and the | |
| 01:31 | vertical line test . So just to kind of break | |
| 01:34 | that down just a little bit more before we solve | |
| 01:36 | these equations , they need to kind of make sure | |
| 01:37 | you understand what I mean by 1 to 1 function | |
| 01:40 | . Exponential function is a 1 to 1 function . | |
| 01:43 | So for instance , you know , it could be | |
| 01:45 | two to the power of X . But it could | |
| 01:47 | be any base in general , any exponential function . | |
| 01:50 | They're always going across the y axis at one here | |
| 01:54 | . They're gonna start , they're gonna go through that | |
| 01:56 | point and then they're gonna go up something like this | |
| 01:59 | . Now , what we basically mean by 1 to | |
| 02:01 | 1 is that for every input these are the inputs | |
| 02:04 | , we're putting into this function , we go up | |
| 02:05 | and we only get one output . So if we | |
| 02:08 | pick for instance , you know this point right here | |
| 02:10 | , this X value could be you know , seven | |
| 02:11 | or something . Then we go off here and then | |
| 02:14 | we intersect and then we read the value off here | |
| 02:16 | and there'll be two to the power of seven . | |
| 02:18 | That's what this is gonna be right here . But | |
| 02:19 | for every input that we give , There's only one | |
| 02:22 | output . In other words , the function doesn't have | |
| 02:25 | any weird wiggles or curves to give you more than | |
| 02:28 | one output is only one input to one output . | |
| 02:31 | And that means it's a 1-1 function . And because | |
| 02:34 | of this property , because of the shape of the | |
| 02:36 | exponential function by the way , even if it went | |
| 02:38 | And flip the other direction , starting high and coming | |
| 02:40 | low . Like we talked about still a 1-1 function | |
| 02:43 | . So four any exponential function with the same base | |
| 02:56 | ? The following . The following thing is true . | |
| 02:59 | Okay . If we have some base could be to | |
| 03:03 | like this thing could be three , could be five | |
| 03:04 | , whatever raised to the power we're gonna call it | |
| 03:07 | a power P . Is equal to b to the | |
| 03:11 | power of Q . That's supposed to be a Q | |
| 03:14 | right there . Um If and only if P is | |
| 03:21 | equal to Q . Right ? So what this means | |
| 03:24 | is that if you're comparing two functions right ? And | |
| 03:27 | to exponential functions with the same base , then P | |
| 03:30 | . B to the power of something is equal to | |
| 03:33 | b to the power of something else . Only if | |
| 03:35 | these exponents are equal . It's a little bit of | |
| 03:38 | a trivial statement , but it's only true because it's | |
| 03:41 | a 1 to 1 function like this . So just | |
| 03:42 | some examples , you're gonna kind of laugh a little | |
| 03:46 | bit when you see these examples . But what I | |
| 03:47 | mean by that is two to the power of three | |
| 03:51 | Is equal to two to the power of three . | |
| 03:53 | You might say . Well of course it is , | |
| 03:54 | it's exactly the same thing . But what this is | |
| 03:56 | basically saying is when you have kind of to exponential | |
| 03:59 | functions with the same base , the only way that | |
| 04:01 | the points on the curves can be the same is | |
| 04:03 | if the exponents are also the same . So if | |
| 04:06 | you have a different exponential function , three to the | |
| 04:08 | power of 16 Is equal to three to the power | |
| 04:11 | of 16 , same base . If you imagine two | |
| 04:13 | different exponential curves , you plug in the value of | |
| 04:16 | 16 , you read the values off these guys are | |
| 04:18 | equal because this is a 1-1 function . All right | |
| 04:21 | . And so in general , if you have a | |
| 04:23 | base of , let's say a nine to the power | |
| 04:25 | of X is equal to the nine to the power | |
| 04:27 | of X . Only because these exponents are equal , | |
| 04:31 | these curves are the same only because these exponents are | |
| 04:34 | equal there . Now we're gonna use this property to | |
| 04:36 | solve . I know it seems a little bit silly | |
| 04:39 | in the beginning , but we're gonna solve some exponential | |
| 04:41 | equations here . So the very first one is very | |
| 04:44 | , very simple . So simple that you could actually | |
| 04:46 | come up with the answer without even doing much . | |
| 04:48 | But let's take a look at this . What if | |
| 04:49 | you have an equation eight to the power of X | |
| 04:52 | is equal to two . Now , for those of | |
| 04:54 | you very clever , you could probably look at this | |
| 04:57 | and you could you could just figure out in your | |
| 04:59 | head probably what the power would be there when you | |
| 05:01 | see the answer , you'll understand . Oh yeah , | |
| 05:03 | that's what it has to be . But for those | |
| 05:06 | of the rest of us like me that can't look | |
| 05:07 | at things and see what the answer is , We | |
| 05:09 | have to do work now what this is saying is | |
| 05:12 | that because the exponential function is 1 to 1 and | |
| 05:15 | because if you have the same base on both sides | |
| 05:18 | of the equal sign , the only way this thing | |
| 05:20 | holds true as inequality is if the exponents are also | |
| 05:23 | equal , Then one way to solve this equation is | |
| 05:26 | to try to make you have an exponential function on | |
| 05:29 | the left hand side and a number on the right | |
| 05:31 | . You want to basically realize that eight to the | |
| 05:33 | power of X is really equal to two to the | |
| 05:36 | power of one . This is another way of writing | |
| 05:38 | what's above the implied one here is I'm just writing | |
| 05:41 | it out for you . If there's a way for | |
| 05:44 | us to make that eight to have a base of | |
| 05:47 | two , then we could equate the left and the | |
| 05:50 | right hand side like this . Think about what , | |
| 05:52 | How can you write eight ? Eight can be written | |
| 05:55 | as two to the power of three . Would you | |
| 05:56 | agree ? Because two times two is four times two | |
| 05:58 | is eight . So I can write that is two | |
| 06:00 | to the power of three to the X . Power | |
| 06:03 | is equal to two to the first . The math | |
| 06:06 | between this and this should be rock solid . All | |
| 06:08 | I've done is taken that base and I've rewritten in | |
| 06:11 | terms of two to the third power . Why did | |
| 06:13 | I rewrite it like that ? Well , it's because | |
| 06:15 | I have a two on the other side of the | |
| 06:16 | equal sign . You'll see in a minute while that's | |
| 06:18 | important . All right . So now I have an | |
| 06:20 | exponent raised to an exponent . So this is equal | |
| 06:22 | to two to the power of three times X is | |
| 06:25 | three X . Is equal to two to the power | |
| 06:28 | of one . Now you see it looks silly right | |
| 06:30 | here when I write it down to to the power | |
| 06:32 | of three is equal to to the power of three | |
| 06:34 | . But now you understand why it's that way , | |
| 06:36 | what it's saying is if I have an expo exponential | |
| 06:39 | on one side with the base of two in this | |
| 06:41 | case and have another exponential on the other side of | |
| 06:43 | the equal side with the same base . Then the | |
| 06:46 | only way these things are equal is if the exponents | |
| 06:49 | are also equal , just like these exponents are equal | |
| 06:52 | and these exponents are equal and these exponents are equal | |
| 06:54 | . If the bases are equal then the only way | |
| 06:56 | the equality can hold as if the exponents are equal | |
| 06:59 | and what that means is I can say right away | |
| 07:02 | then that three X must be equal to one , | |
| 07:05 | we must set the exponents equal and that means that | |
| 07:08 | X must be equal to one third . This is | |
| 07:10 | what you would circle on your paper , X is | |
| 07:13 | equal to one third . So when you were given | |
| 07:15 | the original equation eight to the power of X . | |
| 07:18 | We have to solve for the power of X . | |
| 07:20 | And by solving and going and making these bases the | |
| 07:24 | same across the equal sign and equating the exponent , | |
| 07:26 | we figure out That the exponent must be 1/3 . | |
| 07:30 | Now here in the beginning it's very very important for | |
| 07:33 | us to try to check our work just to make | |
| 07:35 | sure we understand why . So we're gonna take this | |
| 07:38 | answer , we're gonna stick it back in here . | |
| 07:40 | Eight to the power of X . Must be equal | |
| 07:43 | question mark to to . I'm gonna put the value | |
| 07:45 | of one third in here . Equal question mark to | |
| 07:48 | to but what is one third power one third power | |
| 07:50 | is a cube root . So what you're really doing | |
| 07:53 | on the left hand side is a cube root of | |
| 07:55 | eight . And when you do a tree here , | |
| 07:57 | two times four is eight and this is two times | |
| 08:00 | two is eight you're looking for triplets because it's a | |
| 08:02 | cube root and you have a two . So what | |
| 08:04 | you found is that yes , two is equal to | |
| 08:06 | two . So by plugging in an exponent of one | |
| 08:09 | third , taking the cube root , that is the | |
| 08:12 | exponent that satisfies this equation . Remember you always get | |
| 08:15 | an answer bank , you should always be able to | |
| 08:17 | check your answers . It doesn't matter if it's exponential | |
| 08:19 | equations or other kinds of equations that we've solved in | |
| 08:22 | the past . All right . Every one of these | |
| 08:25 | problems is essentially gonna follow the same kind of deal | |
| 08:28 | . We're going to try To make the basis of | |
| 08:32 | the same across the equal sign to set these things | |
| 08:34 | equal . So what if you have 27 to the | |
| 08:37 | power of two times X ? Now notice the exponent | |
| 08:40 | is not just a number or a variable , it | |
| 08:42 | has two times variable . This is still an exponential | |
| 08:45 | function because the variable is in the exponent Is equal | |
| 08:48 | to three . We want to solve for the value | |
| 08:50 | of X . How do we do that ? Well | |
| 08:52 | we recognize that 27 and three are kind of related | |
| 08:56 | . How do you know ? Because if you know | |
| 08:58 | your multiplication tables , you know that 27 can be | |
| 09:00 | written as three to the power of three . How | |
| 09:04 | do you know that ? Because three times three is | |
| 09:07 | nine and then nine times three again is 27 . | |
| 09:09 | So you can write this as three to the power | |
| 09:11 | of three . This is all still raised to the | |
| 09:13 | two X power and it's equal to three . Now | |
| 09:16 | we have an exponent raised to an exponent . So | |
| 09:18 | it's three to the three times two is now six | |
| 09:20 | X . And that's equal to three . And now | |
| 09:23 | that we have the bases across the equal sign the | |
| 09:25 | same . The only way this can be true is | |
| 09:27 | if six X has to be equal to the exponent | |
| 09:29 | over here which is just an implied one . So | |
| 09:32 | X . is the 1 6th power X . is | |
| 09:37 | the 1 6 power . So that's what you would | |
| 09:39 | circle on your exam . Now again we're not gonna | |
| 09:41 | check all of these things but let's go in the | |
| 09:43 | beginning and try to take this guy and plug it | |
| 09:44 | back in here . Here we have 27 to the | |
| 09:47 | two times X . Power X was 16 So it's | |
| 09:51 | two times 16 Equal Question Mark three . So here | |
| 09:56 | you have two times 1/6 . So you have 27 | |
| 09:59 | to over six is just one third . When you | |
| 10:01 | simplify the fraction equal question mark to three . But | |
| 10:04 | you know that the one third power is just a | |
| 10:06 | cube root and you know that 27 can be written | |
| 10:09 | as three times three times three you're looking for triplets | |
| 10:12 | . So then evaluating the cube root , you say | |
| 10:14 | that three is equal to three . Check so that's | |
| 10:17 | why 16 works as a solution again . We're not | |
| 10:20 | gonna check every one of these . I'm just showing | |
| 10:22 | you generally how you would do it . Alright now | |
| 10:24 | I'm going to pause for just a second and say | |
| 10:26 | now that we've done both of these problems . You | |
| 10:28 | see the commonality and what we're doing . We're trying | |
| 10:31 | to look and see if it's possible to raise two | |
| 10:33 | to make the left hand exponential have the same basis | |
| 10:36 | whatever is on the right hand side Here we took | |
| 10:39 | the 27 and we tried to make it have the | |
| 10:41 | same basis what's on the right hand side because when | |
| 10:43 | they have the same bases on on on the both | |
| 10:46 | sides of the equal signs that we can equate the | |
| 10:48 | exponents here . If and only if the exponents are | |
| 10:51 | equal if the bases are also the same . Now | |
| 10:54 | this works for these very special problems , but let's | |
| 10:57 | just say what if the equation was 28 to the | |
| 11:00 | two x . Power . There's no way that I | |
| 11:03 | can take 28 write it as a power of three | |
| 11:06 | . Well , okay , there's not a way to | |
| 11:08 | do it and have whole numbers with exponents which is | |
| 11:10 | what we're trying to do here . So there's not | |
| 11:12 | gonna be a way to say , well if if | |
| 11:14 | this was 28 or if this was 29 if this | |
| 11:16 | was 30 or something like that , I wouldn't be | |
| 11:18 | able to write it as three to the power of | |
| 11:20 | something because three cubes is 27 3 to the fourth | |
| 11:23 | power , something totally , totally different . Right ? | |
| 11:26 | So because it's 27 I'm able to do it but | |
| 11:29 | if it was 29 or 40 or 42 or something | |
| 11:31 | else , I wouldn't be able to do it like | |
| 11:33 | this . So these solution techniques are kind of for | |
| 11:35 | the basic problems because they're set up later on , | |
| 11:38 | we're gonna learn about the concept of the law algorithm | |
| 11:41 | . Al algorithm is kind of the opposite function is | |
| 11:44 | called the inverse function of an exponential and a log | |
| 11:47 | a rhythm . Once we learn what they are is | |
| 11:49 | going to allow us to solve any of these exponential | |
| 11:51 | equations no matter if we can write the basis , | |
| 11:54 | you know , perfectly like we're doing here . But | |
| 11:56 | for now we're gonna not gonna worry about that , | |
| 11:59 | we're gonna learn how to solve these problems where the | |
| 12:00 | bases are chosen carefully to make it easy for us | |
| 12:04 | . I shouldn't say to make it easy for us | |
| 12:06 | , but just a certain subclass of problems that we | |
| 12:08 | can solve a little bit easier . What if we | |
| 12:11 | have eight to the power of X is equal to | |
| 12:14 | 1/4 and we want to solve this , we're gonna | |
| 12:15 | pick up the pace a little bit . Now people | |
| 12:18 | get confused when you have this uh this fraction over | |
| 12:20 | here . So let's just try to work with 11 | |
| 12:22 | side of the equal sign at a time . Okay | |
| 12:25 | , 1/4 is one over . You can write that | |
| 12:27 | as two squared . And you know that 1/2 squared | |
| 12:31 | can also be written as two to the power of | |
| 12:34 | -2 . So now you see have transformed the problem | |
| 12:37 | from this to this . These are exactly the same | |
| 12:40 | thing and what you want is a base on the | |
| 12:42 | same side of both sides of equal sign of the | |
| 12:44 | same to be the same number in this case too | |
| 12:47 | . Because we know that eight can be also be | |
| 12:48 | written as two to the power of three , still | |
| 12:51 | all raised to the X . And two to the | |
| 12:53 | power of negative two . So now that we have | |
| 12:55 | this we can write this as two to the power | |
| 12:57 | of three X . Multiplying the exponents together is to | |
| 13:00 | to the negative to you see how we went from | |
| 13:02 | this which looks completely unrelated down to where the bases | |
| 13:06 | are exactly the same here and now we can just | |
| 13:08 | do the final nail in the coffin here and say | |
| 13:11 | that the exponent three X must be equal to negative | |
| 13:13 | two . So X must be equal to negative two | |
| 13:16 | , divided by three . Getting this guy by itself | |
| 13:19 | and you have negative two thirds for the power . | |
| 13:21 | And I promise you , if you take this and | |
| 13:23 | put it in here and then you're gonna have to | |
| 13:25 | remember how we evaluate exponents , where the fractional expense | |
| 13:28 | you have to raise to the power of to take | |
| 13:30 | the cube root , that kind of thing . In | |
| 13:31 | fact , you can kind of see it . If | |
| 13:33 | you put it in here , you can take the | |
| 13:35 | well , you know what ? I don't want to | |
| 13:36 | do it in my head . I'd rather you get | |
| 13:38 | a sheet of paper in that way we don't lose | |
| 13:39 | anybody . But if you take that exponent in and | |
| 13:41 | go through and do the steps that we learned with | |
| 13:45 | rational exponents , how to evaluate this as an expert | |
| 13:48 | , you're going to find the answer is 1/4 right | |
| 13:50 | ? So I encourage you go off and do that | |
| 13:52 | . All right . All right , let's go ahead | |
| 13:54 | and crank it up and go off to the next | |
| 13:57 | problem . What if we have 3 to the power | |
| 14:01 | of X is 1/27 . All right . Same kind | |
| 14:06 | of thing . 27 . We know how we can | |
| 14:07 | write that in terms of a base of three . | |
| 14:09 | It's going to be three to the power of three | |
| 14:12 | , but then we can bring this upstairs and make | |
| 14:13 | it three to the power of X is three to | |
| 14:15 | the power of negative three . And now the basis | |
| 14:18 | of the same could just simply say that X is | |
| 14:20 | equal to negative three and you don't have to do | |
| 14:21 | anything else . All you were trying to do is | |
| 14:22 | figure out what X was equal to and you have | |
| 14:25 | your answer . All right . Yeah . Now this | |
| 14:29 | one stumps people in the beginning , but it's not | |
| 14:32 | any harder . What if we have eight to the | |
| 14:34 | power of two plus X . Is equal to two | |
| 14:37 | . Now it might look a little weird for your | |
| 14:39 | expanded to have a plus sign in it . But | |
| 14:41 | trust me , exponents can have anything in there when | |
| 14:44 | you get down the road and math you'll find that | |
| 14:45 | exponents can have entire equation almost like a long giant | |
| 14:49 | expressions with parentheses and all kinds of things . Things | |
| 14:52 | up in the exponents . When you get to calculus | |
| 14:54 | you'll figure out that exponents can have integration and all | |
| 14:58 | kinds of other advanced math will learn later . You | |
| 15:00 | can have all that stuff inside of an exponent . | |
| 15:02 | So a little plus sign . Yes , it looks | |
| 15:05 | a little weird at first but just kind of get | |
| 15:06 | used to the idea that that can that can happen | |
| 15:09 | . Now we're gonna write the eight is two to | |
| 15:10 | the power of three . This is still raised to | |
| 15:13 | the two plus X . Power . And now we | |
| 15:15 | have to multiply these but you're saying three times the | |
| 15:19 | expression two plus X . You have to distribute it | |
| 15:22 | in . So what you're gonna get is six plus | |
| 15:25 | two X . Make sure you understand because it makes | |
| 15:28 | me if I want to write this explicitly , I | |
| 15:30 | think I do . Instead of writing it like this | |
| 15:32 | just to make sure we don't lose anybody . We're | |
| 15:34 | gonna multiply this out as three times two plus X | |
| 15:38 | . And then we're gonna multiply it through . So | |
| 15:39 | we're gonna get to to the three . I'm sorry | |
| 15:42 | six plus three X . Six plus three X . | |
| 15:47 | Is equal to to the bases are now the same | |
| 15:51 | . And so we equate the exponent . The exponent | |
| 15:53 | is this entire thing six plus three X . Is | |
| 15:58 | equal to the exponent over here . Which is just | |
| 15:59 | one . When we subtract the six we're gonna get | |
| 16:02 | three X . Is negative five and we divide by | |
| 16:06 | the three . We're gonna get X . Is negative | |
| 16:07 | five thirds . So we ain't negative 5/3 . This | |
| 16:11 | is the final answer . If you take this exponent | |
| 16:13 | of negative five thirds and you put it in here | |
| 16:15 | and you add two to it . Simplify the fraction | |
| 16:17 | and then do the rational exponent business with eight raised | |
| 16:20 | to that power . You're gonna find that the answer | |
| 16:22 | comes out exactly equal to two . All right , | |
| 16:26 | one last problem in this lesson , we'll get more | |
| 16:30 | practice in the next lesson . What if we have | |
| 16:32 | 27 to the two X -1 is equal to three | |
| 16:39 | again . Don't be worried so much about the fact | |
| 16:40 | that there's multiplication and subtraction going on in that experience | |
| 16:44 | . But we see that 27 and three can be | |
| 16:46 | written as the same base three to the power of | |
| 16:49 | three is 27 and the expanded is two X -1 | |
| 16:53 | . When we multiply these together because it's an exponent | |
| 16:56 | raised to an exponent it's three times 2 X -1 | |
| 17:00 | . Like this Is equal to three . We multiply | |
| 17:04 | it through . What we get is six X minus | |
| 17:06 | three multiply . Here's six X . Multiply here minus | |
| 17:09 | three . Now bases are now the same . We | |
| 17:11 | set this X moment six X minus three equal to | |
| 17:14 | this X moment like this . And then what we | |
| 17:18 | do is we add the three so we get six | |
| 17:22 | X is equal to 43 plus the one or one | |
| 17:25 | plus 23 and then X . Is four divided by | |
| 17:29 | six . Simplify the fraction divided by two and divide | |
| 17:33 | by two you get two thirds that's the final answer | |
| 17:36 | . Alright so here in this lesson we have learned | |
| 17:38 | how to solve these elementary um exponential equations . But | |
| 17:43 | you see in every single one of these problems here | |
| 17:46 | we change the problem so we had a base of | |
| 17:48 | three on both sides here we change the problem have | |
| 17:50 | a base of three on both sides . Here we | |
| 17:52 | change the problem to have a base of two on | |
| 17:53 | both sides and that's how every one of these were | |
| 17:56 | solved . What we want to do is learn and | |
| 17:58 | get comfortable with this process and the next lesson I | |
| 18:00 | want you to follow me onto there so that you | |
| 18:01 | can again get practiced with these types of problems a | |
| 18:04 | little bit more complicated and just keep in mind when | |
| 18:06 | we get past the next few lessons and cover the | |
| 18:08 | concept of a log a rhythm . We're gonna be | |
| 18:10 | able to solve all of these equations and more complicated | |
| 18:14 | equations that our exponential equations with a more powerful technique | |
| 18:18 | . Once we learn the concept of a law algorithm | |
| 18:19 | , so make sure you can follow me and solve | |
| 18:21 | all of these problems , Follow me on to the | |
| 18:23 | next lesson and will continue right now . |
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Quality Math And Science Videos that feature step-by-step example problems!
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08 - Solving Exponential Equations - Part 1 - Solve for the Exponent is a free educational video by Math and Science.
This page not only allows students and teachers view 08 - Solving Exponential Equations - Part 1 - Solve for the Exponent videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.
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