A Level Maths Tuition Lesson 10 Exponentials and Logarithms Exam Practice!

hey guys and welcome back to another a-level maths tuition video So today we're going to be covering exponentials and logarithms been a long time coming so with exponentials you should hopefully be aware of Powers right so indices and things like that so the graph those of you that are interested for all power laws is exactly like this this is y equals a to the x and it always crosses out one because uh any number to power of zero gives you one and exponential is just where a is equal to two uh equal to e which is equal to 2.7 dot dot okay that's the only difference and the reason for this number is that it turns out in a lot of kind of calculations that we do especially in things like physics and economics and it turns out that the bass tends to be a number which is 2.714 and so on now manipulating this should be relatively easy right so if you remember your rules of indices which I hope you do so for example you have a to have M times a to the power of n you just add the powers and slow on I'm not going to go through these today but we will go through the opposite so one thing that you might not know is that there is actually an inverse operation which isn't just taking the square root and it's called a logarithm so a logarithm all it is is let's say you have log to the base 10 of a thousand what this will tell you is how many times do times 10 by itself to get a thousand and the answer will be 3 right 10 times 10 times 10. and if you look on your calculator here's where the button is it's just over here and you'd fill that in so 10 and a thousand right you get three you can also say for example have a different base so you could have log to the base 2 of 8. so how many times do I times 2 by itself to get eight well that should also be three okay so again we'll just go through and we'll change these all three eight even and you get three okay so this might not seem too useful at the moment because surely you could just kind of take a square root or something like that however we should keep in mind that we don't always have numbers that are super easy so for example what would the log of base two the number nine be but this is where things can get a bit more interesting and then of course we have algebra so logarithms have all of their own rules associated with them and here are some of them so uh before we get started actually what I will say is that generally speaking if we have logged the base 10 we don't bother writing the 10. so if you ever see log 100 that is the same as writing log to the base 10 of 100. when we work with x exponentials you're going to be tempted to have log to the base e of e squared or whatever that is the same as writing Ln of e squared okay so that's just so you know for now and again the most important aspect of this is going to be um things such as rearranging algebraic expressions it's going to come up in integration and differentiation and if any of you do kind of physics or economics you have logarithmic graphs as well that you'll need to do so it is quite an important uh aspect of maths even if it doesn't seem too useful at the moment so let's look at the rules now the rules here are not too bad so log to the base B so again B is just going to stand for base it has to be the same of the letter let's choose X plus log to the base B so again same base Y is the same as doing log to the base B of x times y okay so if you add logarithms together you can multiply them now B keep in mind that in your specification you have to use these rules both ways okay so sometimes you may have log of let's say eight and you may be expected to do log of 2 plus log of 4 instead and vice versa so you might be able to guess that if you have log B of Y minus log be of X then you're actually going to be dividing them and again you need to be able to do this both ways the next kind of things that you need to do is the power law which just states that if you've logged the base B of x to the power of Y that is the same as just moving the Y to the front and having log b x this is one of the most important rules that we're going to come across it comes up a fair amount now another thing to note is that the logarithm function is the inverse function of power so it's going to be an inverse graph so if you were to draw it on here it's going to be like this where it should be a mirror reflection so this is y equals log X and this value here is always going to be 1. so what that means is that log of 1 and it doesn't matter what base you have always equals zero okay and that's what the same reason is as this anything to the power of zero gives you one so vice versa anything to the um log of one is always going to give you zero and likewise if you have log of a base of a base that should give you one and this should hopefully make sense right like if I give you an example if I say log of 2 2 I'm asking how many times do I times 2 by itself to give two well just once and that's going to be true for any pair of numbers right so that should hopefully be okay the last one um actually a couple more that we're going to do B of B to the power of K so if I take the log to the base of a power of a power then I just get rid of the power so this is going to be really important when we come to rearranging equations okay so what you'll find is you may have let's say um 3 to the power of x then what you can do is apply log to the base 3 of this whole thing and just get X and you do that to both sides and that can make things a lot easier for you okay and lastly in a similar vein if you have base B log base B of K it gets rid of it so you can always do the inverse so if I apply log I get rid of a power if I apply power I can get rid of a log okay so this is pretty damn complicated um this might seem like a lot of stuff but as we go through it it's going to be nice and easy and will make sense okay the other thing to note is that these numbers must be positive okay so in all of these cases the number inside must be positive you can't take log of a negative number so let's get down to doing some of these questions so let's stop this um something that should be a relatively easy and we have 3x well 3 to the power of x equals 13 and we have to find the um answer so using what we had before if you remember we can apply log to the base of that power so if we ever want to get rid of a power which we do in this case all we do is we apply log to the base of that power equals log 3 13. so just to give you an example or a reason as to why this works if you remember from the power law right this one here we can bring the power to the front so now we have X I'll only do this once okay so don't worry I won't do this every single time because this is meant to be fairly quick now what did I say look to the base of a number oven of the same number is always one so how many times does three multiply by itself to get three it's one so this is going to be X is equal to log base 3 of 13. so you can jump to that step straight away you don't need to do the intermediate ones because it is a rule but now you just need to use a calculator three gives me 2.33 to three significant figures not too bad hopefully so again these will gradually get more tricky so again we have 2 to the power of x equals 32 so we want to take the log base 2 of both sides because again we always choose the base of the power to get rid of the power 2 to the power of x equals 32 and just like in the previous question that's going to give me x equals my apologies I forgot to do the log so do that vlog two again you can use a calculator but hopefully you should know this one again so these are just quick two more questions quick rapid fire but they do come up quite a bit in exams now this is where things get a bit more interesting slightly more interesting we need to solve an equation okay so what we need to remember here is that we have two separate logs now some people mess this up quite a bit so you can't just subtract these straight away because there's a 2 in front so just so you're aware you cannot do that's not the correct thing because this two when we talk about logarithms represents a power okay so we need to keep that in mind so we need to use our rules in Reverse now so this will become log to x squared again remember the power rule from the beginning you just just in reverse minus log 2 5 and that still equals one so now we can use the subtraction rule which says if you subtract two logarithms all you do is you divide them okay now how do we get rid of logarithms well in this previous part we applied a logarithm to get rid of a power so if you want to get rid of a logarithm you need to apply a power so what I like to call this is Raising to the base okay now I'm going to do it fully once okay so all I'm going to do is write 2 and then there's going to be to the power of log to the base two x squared of five okay now I'm only going to do this once because after this you should know and again you do that to both sides you raise everything to the power of two well the base 2 raise everything to its own power okay now I'm only going to see this once if you remember from our rules up here B to the power of log to the base B just gets rid of the log and the power okay so in this case we have 2 to the power of a base two so if you like you can cancel that through and you're just left with x squared over five equals two to the power of one which is two okay so I'll only do that once but in general if you just raise something to its own power so a logarithm to its own base it gets rid of the logarithm okay so now it's just a simple case of x squared equals 10 x is equals plus or minus square root of 10. okay which it doesn't it doesn't say to give any kind of decimal place or anything but that should be it cool so again solve the equation um keep in mind you again you can't do this unless you combine the logarithms okay so we're adding them which means we will times what's inside so we have log to the base three x times 4 gives me 4X equals 2 and again because we want to get rid of a log to the base 3 we're raising everything to the power of three I know the way I'm saying that doesn't sound right I'm not cubing everything I'm making 3 the base of everything I'm putting that to the power so on the left hand side that's just going to get rid of the log on the right hand side that's just going to be 3 squared okay so the 4X equals nine x equals 9 over 4. so again these are four Mark questions really really short really really short they shouldn't provide too much hassle all right let's keep going they do get harder and harder and harder Express as a single logarithm to the base a h so again this shouldn't be too bad this is like part of um question three first things first I can't combine unless I get rid of the power first and the number in front of the lock so let's bring it in cool now I just subtract them so I divide them so this is log is a technically I didn't do it quite accurately this would be X Plus 1 squared not the whole logarithm squared that's okay super easy again the questions do get harder as you go along so for example simultaneous equations okay this is going to be good fun um might be a bit tricky but what we need to do is try and find equations or expressions for both of these so we can solve it so the problem with simultaneous equations is we either need to sub X into here or sub y into there either way it's fine now I don't think this is too bad but some people so okay there are two ways to go about doing this right so the first way is if I call this equation one equation two with simultaneous equations there are so many different solutions to this but X would just equal e to the 2y minus 1 right now what I could do right is change X all right I can now sub this in to this however what might make my life a bit easier is if I make it look like Ln x minus 2. so that's minus one from both sides uh minus two from both sides even I can count x minus two equals e to the two y minus three now if I take the natural log of both sides right I will have Ln of x minus 2 equals Ln e to the 2y minus three and that is another way to do things the other way to do it is to Simply raise all of this to the path e and let that cancel out okay either way is again completely fine so let's have a think about what we want to do so let's take the natural log so Ln x minus 2 equals Ln e to the 2y minus 3. cool and if you're wondering why I haven't like done the um canceling things so I haven't canceled Ln and Easter twice because I'm taking Ln of the whole thing okay so now I can sub this into here so we have Ln e to the 2y minus 3 equals 2y minus 1. okay so we can do a few things to cancel this out I could raise everything to the power of E which will get rid of log to the base e so we have e to the 2y minus 3 equals e to the 2y minus 1. now this is where things might get tricky okay e to the 2y minus 1. if you remember your laws of powers is the same as doing e to the 2i multiplied by e to the minus 1 right because then you add the powers together okay now if actually I will e to the minus one is it's one over e so this is equal to e to the 2y multiplied by 1 over e so in other words it's e to the 2y over e it says write down e to the 2y minus 3 equals e to the 2y over e okay so uh that's meant to be a y so what we can do now is move all of the Y's to one side and all the numbers to the other side that's right or the E's through one side so if we move this to the other side this to the other side the e to the 2y minus E to the 2y over e equals three factorize out in e to the 2y and move that over to the other side or divide it so e to the 2y and all we need to do now is take the log of both sides so 2y and yes this looks absolutely heinous it's totally fine remember e is just a number so that should be totally fine let's get our calculator out so 0.5 Ln is this button here 3 over so 1 minus e whoops that's not even right that's 1 over Alpha e and yeah 0.7786 so does it tell me what how many decimal places uh to two decimal places that'd be 0.78 okay so again this is a seven markers so we can't really expect the answers to be um nice at all so what I'm going to do is just in the interest of room is we need something to find what x is uh the easiest X equation I can find is equation one right x equals two y minus one so x equals e to the 2 times what we just found out 0.78 minus one but I'm going to use the answer button of course to make sure my answer is perfectly accurate so we're going to do is do whoops if the power is shift Ln here to answer button and then minus one which gives me an answer of 3.75 to two decimal places and that's a seven mile question um it's honestly it's not that bad um if you think about it it's ugly sure but it doesn't mean it's difficult okay but uh yeah I think it's I think it's okay seven marks is a lot of marks it's probably one of the highest marked questions that you'll get okay something that's slightly uh easier to digest we have Ln 2x plus 5 equals one we have to solve for x okay so we have Ln 2x Plus 5. 1 and again the opposite of Ln is Raising everything to the power of E okay so this that will get rid of the Ln equals e to the power of one okay so then all we need to do is rearrange this so X is equal to e to the one minus five over two and that will give us an answer because it's an actual number now you are allowed to leave your answers in terms of e because e is an irrational number it goes on for an incredibly long time so this would give you full marks but if you want to be super accurate I guess not accurate but if you want to write it out to that I like to write it in both ways and that way the examiner knows I know what I'm talking about and they can give me marks for either one either way totally fine so next question this is where things are going to get a bit tricky because we have some algebra and we have some substitutions and things like that so hopefully this is okay so far so we have y equals log to the base to X so if we have a look at the first one which is log to x squared the most common thing I see is y squared which is not right look what it is the power law means that this would become 2 log to the base 2 of x so if Y is equal to log base 2 of X then that is equal to 2 y okay just keep that in mind okay now let's go for the next one part B uh this one some of the things I see are just blank looks of confusion now the thing is we can't just write Y in terms of this in fact we can't really change that into this in any easy way so let's have a think how can we make log 2x out of this well if we use the addition rule right this is 2 times x so this is the same as writing log to the base two of two plus log to the base 2 of X Now log to the base 2 of 2 is 1 and log space two of X is y done this next one is actually not on a lot of your specs anymore it's a slightly older question so log to the base 8 of X this is called a change of Base so we need to do change base and what that means is I'll show you the log to the base a of X is equal to log to the base 2 of X because 2 is the base we want over log to the base 2 of that so essentially if you want to write this out log to the base B of A is equal to log to the base C of a right so C is the new base over log to the base um of the new base of the old base okay and of course that'll make y logs the base two of eight is obviously three because two cubed is eight okay and that'll be two marks a pop six marks in total not bad at all so again the change of bass hardly ever comes up but it's a good thing to remember if it's on your spec so question nine here you go in exact form means that we're probably going to have um Ln or E you know answer in this case it's going to be Ln but in other cases it could have been e so this is going to be kind of interesting so what do we want to do two ey let's move everything to one side first of all now this kind of reminds me of a quadratic equation the only problem is is this isn't squared and this isn't to a normal Power so what we're going to do is we're going to multiply through by e to the Y so that gives me 2 e to the 2y right and that's going to give me 15 because e is the minus y times e to the Y that will give me e to the 0 which is one minus 11. so if I write this out like here that looks like a quadratic uh one way I could make it look more like a quadratic is say x is equal to e to the e to the Y and that case I'll have two x squared minus 11x plus 15 equals zero and now what you're able to do is factorize completing the square quadratic formula whatever way you want to do it um but if we factorize this it's just going to be 2 e to the Y and in the other bracket have to be e to the Y or again if you want to look at that over here it's going to be 2X and x and then we're going to have -5 and -3 zero so if I flip over to here it's going to be minus 5 and minus 3 equals zero really doesn't matter each way so either we have 2 e to the Y minus 5 equals zero or e to the Y minus 3 equals zero and we solve it in the exact same way so we have e to the Y equals five over two so take the natural log of both sides we get Ln pi over two again it can't be less than one just bearing that in mind or y equals so I should say e to y equals three and then okay so again if one of them is negative we'd be able to Discount that immediately which can be a useful thing to do okay here we go so this is part of the exponentials and logarithms uh chapter is they really really really have a big focus on what I like to call uh synoptic maths so this is where matters apply to real world situations um bearing in mind there is one question you may not be able to do yet but I'm going to go through it even though we haven't gone through it in one of these YouTube videos because this is going to involve differentiation okay and that's because it says rate anytime we have a rate it's going to be differentiation okay but I'll go through it I'll talk through it but it's this isn't a differentiation video okay so population and this by the way this kind of question comes up every single year right and the structure is identical they will always ask your initial number then they'll ask you a rate then they'll ask you to work out how long it takes this question and this style comes up every single year okay because it's the perfect synoptic question it teaches you about exponentials it teaches you about differentiation it teaches you about exponentials again okay so this question is basically guaranteed to come up and it's the same exact formula okay initial rate of increase how long that is it always always always always Part D is sometimes like a limitation so some kind of limitation okay in this question it doesn't have it but in some questions it will have it but everything else is the same species of a population of a species of plant model using the formula P equals 50 e 0.1 T which is the number of weeks okay cool if I'd done the number of plants when the population was first recorded if it's first recorded it means T is equal to zero no weeks have passed so p is equal to 50 e 0.1 times 0 is 0 so this is going to be 50. okay now here's the trick to this question whenever it says initial it's always just going to be the number in front of E okay so this one question it's just the number in front of E Part B says find the rate of increase 10 weeks after the population was first recorded the first thing we need to do is find the rate of increase so if P equals 50 e to the 0.1 T then we need to find DP by DT so to differentiate so if we have um y equals e to the um what should we call it ax then d y by DX is going to be a e to the ax so all you do is you bring the thing forward so here I'm bringing 50 uh 0.1 forward so we have 50 0.1 Times by the exact same thing again okay so 50 times 0.1 just gives me 5 e to the 0.1 t Okay differentiating exponential is super easy now it says 10 weeks after the population was first recorded So when T equals 10 DP by DT equals 5 e to the 0.1 times 10 so in other words it's 5 e to the 1 which is going to be um I actually have to use a calculator for that because it's going to be 5 times e so five thirteen point five nine so does it say to 3sf nope but we should probably give it to 3sf just because it's a long number that's it two marks okay now part C is where things get a bit funky so we're looking for find out how many weeks it takes for the number of planets to exceed 300 so it's subbing in P equals 300 and again this structure is always the same initial amount T equals zero B you're going to differentiate C you're going to sub in something for the um P or whatever it is it's always the same okay so you have 300 equals 50. e to the 0.1 T okay this is four marks it's not too bad we want to rearrange for T so let's divide both sides by 50. remember only take logarithms if you have one number in the expression right I if I take logarithms of this at the beginning it's going to mess everything up only have one number one exponential that's when you take logarithms okay likewise if you want to take the exponential of both sides right you want one number one exponential one logarithm that's it so 300 divided by 50 is 6 equals e to 0.1 T now I have one number and one exponential so I'm going to apply the natural logarithm to both sides so ln6 equals 0.1 T So T is equal to times both sides by 10 or whatever you want to do was that you can divide both sides by 0.1 as well if you want so it's going to be 10 natural log of 6 that gives me an answer of 17.9 blah blah blah blah blah right now if you look at the question it says find out how many weeks it takes for it to exceed so it can't take 17.9 weeks right it's how many whole number weeks so it takes always round up 18 weeks and what I mean by always Round Up is if this number was 17.0001 week s you still round to 18. because it's taking 17 weeks and a bit so how many total weeks does it take it takes the next number up there you go seven marks not too tricky hopefully not too bad so for those of you doing a level physics you're going to feel right at home here for those of you that aren't still the same stuff the care of a radioactive substance is modeled using the formula n equals a thousand e to the minus KT right in look at this right well n is the number of atoms called write down the number of atoms when it started to Decay well again we're subbing in t equals zero so n equals just the number in front of E okay that was hard given that it takes 14.4 years for half of the substance to Decay find the value of K to 3 SF so in other words when T is equal to 14.4 and again remember it's always a substitution right now I may have led you to believe that it's always the same but keep in mind there's always 10 unseen marks so this is where Part B was the Unseen Mark or unseen question of the paper okay it's always substitution but they've changed it now slightly okay but normally it's always kind of same thing right n is half so 500. so let's sub that in 500 equals one thousand e to the minus 14.4 k okay shouldn't be too bad divide both sides by a thousand I get a half equals e to the minus 14.4 okay so again remember we want one number one exponential now we take the natural log of both sides add another half equals minus 14.4 K if you're worried about getting a negative right just so you know you can't Ln log less than zero but if the number is between zero and one log of n is always less than zero it's negative now think about why to get Ln of one over two I need to take the reciprocal so I need to have a negative power so if you have log of a number that's less than one but bigger than zero it's okay for this to be negative okay so I know some the reason why I'm going through this right is obviously when you put it in a calculator you're going to realize that it's completely fine the problem is that some students panic when they say they've got a negative they're like oh man this is going to be negative that's not right it's right it's fine so just make sure that you're being super aware of that okay another 0.5 instead of uh one over two because I'm lazy and uh 14.4 and look it's a positive number 3sf that'll be 0.0 481 remember we only start counting from the first non-zero number and then lastly part C I remember it being fairly easy calculate the number of atoms left when T equals 30. easy so n equals a thousand e now we know what K is Right K is 0.0481 and T in this case is T equals 30. so all we're doing is subbing numbers into a calculator for a mark right a-level maths so let's click call so we have 1 000 and then we have shift e to the minus 0.0481 Open Bracket 30 close bracket 236 and well hang on actually wait okay so some students actually would leave this as 236.2 how many atoms are left can you have 0.2 of an atom nope so it's 236 atoms okay it's an easy question but sometimes just sub it in they round it to two decimal place or something some students will be lucky and we'll round to three significant figures and just happen to get the answer right but obviously you don't want to rely on luck or happenstance be super careful with that okay so again these are the kinds of questions you can expect and again we have another kind of question and rate of change etc etc so this is probably one of the hardest style of questions because again it was unseen so let's go through it if you can do this question you can probably do I would say pretty much do any a level question uh it's all I'll have time for today after this question but again when we start going through exam papers we will encounter far more and um honestly they all are very similar but there will be some variation in that sense where obviously the equation is different so it's good practice find the rate of change of the temperature in terms of K cool rate of change means we're doing differentiation so again what we're going to do is bring the number in front of T forward so minus k and then the rest of it stays the same now constants when you take the rate of change go away so all you have here is minus 75 okay e to the minus KT now if we think about this right where the temperature that's right this is the temperature after turning the kettle off we're saying how quickly does the temperature change it should make sense it's negative right because the temperature should be decreasing over time that's two marks you know that's not really bad Part B after five minutes the temperature of the water is 70 degrees so what we're saying is 70 degrees equals 75 e to the minus uh how many minutes was it sorry five okay plus 22. also this plus 22 should also make sense if you think about it that's room temperature right the temperature of the water should never go below room temperature right so these maths equations like they do put some thought into it they don't pull numbers out of there of thinner okay so minus 22 to both sides that gives me 48 equals 75 e to the minus 5K divide both sides by 75 was e to the minus 5K oh take natural log both sides we have Ln and again the reason why it's the natural log is because my power is e okay so we're taking log to the base e which is Ln and again there's a negative but that's a good thing because this fraction is less than one so we have Ln 48 over 75 over -5 will give me the answer it's going to be a bit slightly painful to type into the calculator so we have Ln and then we have a fraction 48 over 75 and that's over minus five we get 0.0893 uh it doesn't say how many sig fig but it's normally three is good so 0.08 93 if you're wondering why these numbers are so ugly it's because again these equations they don't come out of thin air uh K and these constants tend to be very small or very weird numbers right the universe doesn't really play by our rules in that sense in terms of having whole numbers and things like that we're not normally lucky but how many minutes it takes to cool to 55 degrees so uh T equals 55. so we have 55 equals 75 I'm just doing this based on memory so I will check in a second t so that's what our equation looks like good subtract 22 we get 33 . T and again we're doing basically the same thing we just did uh 33 over 75 equals a it's minus 0.0893 t natural log both sides T and then lastly divide both sides and because I'm an absolute genius I've actually left this as is so what we can do is actually edit edit the equation sort of writing the whole damn thing again 9.193 blah blah blah blah I remember what I said how many minutes does it take Well it can't take 0.19 of a minute so it takes 10 . okay so I know that logarithms and exponentials are slightly obscure things right they're very difficult to imagine so I thought it might be good to go through some real life applications for them um so again if you're doing physics exponential exponentials and logarithms anything to do with physics or engineering Etc even if you're going to become a doctor for example you're going to have exponential decay of things like um medicine that stays in the blood for example um and things like that you also have exponential when you're talking about like the stock market for example right uh wealth in the world tends to increase exponentially so those are the kinds of things you can think about compound interest literally anything with powers has logarithms as well so they're very useful things to talk about indeed but that is a pretty good summary of all the different types of questions you can be asked and uh that should be enough for today so I hope you enjoyed