A Level Mastery - 10 mark Logarithm and Exponential Question! SOLVED

hey guys and welcome back to the a Mastery Series this time we're here with an exponential SL logarithm question now the reason why uh this topic is kind of brought up into one is because you can't actually have logarithms without exponentials so in every single question I've ever seen ever they always put some kind of uh question where you have to rearrange something so we have an exponential here but I'm willing to bet that we're going to be using a logarithm likewise if it was a logarithm question they always put a part where you need to work out the exponent okay so if we have a look at this question we have an equation here that has everything that's unknown but our two variables are H and T so we have a B and K which are unknown constants that we are unaware of so when a tree of a certain species was planted it was 2 m in height so what I'm going to do here is as I go through I like to highlight the information so when it was planted means when T equals z because it says T is the years after it was planted so when it's first planted of course there's no time that's passed then when tal 0 we have h = 2 and after 2 years it height was measured at 3.81 so that means 2 H = 3.81 just like that okay so we've already got some information that's quite useful the height H meters of this tree T years after it was planted is modeled by the equation h = a subtract b eus k t where A and K are positive constants Co given that the species of tree will reach in its lifetime a maximum height of 12 M find the value of T when H equal 10 cool so as you can see this is what I mean by using logarithms right because T is in the exponent over here we're going to use logs to rearrange that so of course the problem here is that we don't know what a B and K are so I can't just Sub in h equal 10 and then try and work out T because I'll get a bunch of algebra so I need to work out a B and K first which is exactly why this is 10 marks now if I sub in tal 0 into this equation remember that means this exponent here this exponential will become one which means I'll have two unknowns still I'll have 2al a subtract B so I can't use this first one just yet the second one well it's going to be the same issue except even worse because we'll have K as well so we can't use these two pieces of information yet because they'll provide me equations that have more than one unknown but there is another piece of information given that this species of tree will reach in its lifetime a maximum height of 12 M so what does that mean well if we're talking about in its entire lifetime you might be thinking well how do I know what the lifetime of a tree is well you don't need to know the idea is that the maximum height of the tree is actually just going to be a think about it as time goes on if you think about the exponential uh negative exponential function if I I draw it out here it's going like that it's an exponential decrease so initially we are subtracting the biggest possible amount from a when it's first grown so the height is going to be very very small because we're subtracting the biggest possible number and as time goes on the amount that we're subtracting exponentially decreases if you'd like you can even sub into a calculator right e to^ of Zer gives me one but e to the power of anything that's negative so let's say minus one has to be smaller than one and as it gets bigger so if we do it to minus 10 now as you can see the numberers getting smaller and smaller and smaller so in other words all I'm trying to do is see what happens as T goes to Infinity because remember since this is infinitely decaying no matter how long the tree lives the idea is this term here is going to go to zero it's going to disappear okay then H is is just going to equal a because as we increase that value of e this whole term is getting smaller and smaller and smaller 10us 44 and it's getting there faster and faster and faster so that's how we use that piece of information it says it reach a maximum height of of 12 M so that means 12 is equal to a now this actually makes sense in terms of the structure of the equation if you think about it they're trying to use this equation to show what the height of the tree as T is after it's planted we can see that as T increases this thing here gets smaller which means we're subtracting a smaller number so it makes sense that a is actually the starting or the maximum value for the height so you don't actually need to do this maths you could have deduced by inspection that a is going to equal the maximum height because I mean hopefully that makes a bit more sense but even if you didn't understand or can't logically inspect this equation to figure out that a is the maximum height you can just sub in a good old fashion value by seeing that t as it increases makes e exponentially decrease perfect now if you remember rightly this equation here originally had two unknowns but now we know what one of the unknowns is so we're going to use this expression now so when tal 0 h = 2 if we sub that in we have 2 = 12us B why because e^ 0 = 1 so that's going to be B * 1 there we go so swap that around we're going to get B is equal to 10 and now finally so we've used this last information this piece of information we're going to use this piece of information to find K so when T = 2 h = 3.81 from here it's going to be relatively easy 81 um it might be good to rearrange the equation first which um might should we do that let's rearrange the equation first so at the moment we have H equals 12 subtract 10 eus KT so let's rearrange for K because that's basically what we want to work out in this case so let's subtract 12 from both sides we have hus 12 actually I'm going to do it the other way around I'm going to move this to the left side and this to the right side so that gives me 10 eus K T = 12 - H so then eus KT = 12 - H over 10 take the natural log of both sides we get KT equals Ln 12 - h/ 10 so then t no we're not looking for T we're looking for k 10 so now we can sub in all our values and that makes our life hell of a lot easier now again remember the purpose of these videos is to show you my thought process and how to get an A star so it really is about right kind of clearly labeling the information that you have and then thinking ahead so there are students that would have subbed this in realized they can't do anything and then paniced sub this in realize that that's even worse and they get confused so it's good to think through what you want to do first by labeling the information and then imagining what happens that's all I've done in this case 12 - 3.81 / 10 over -2 so let's quickly sub this all in so we have the natural logarithm of 12 subtract 3 81 over whoops over 10 and then divide by- 2 gives me 0.998 and then some other stuff we will leave it there to uh four decimal places or three SigFig now finally we can work out the value of T when H is equal to 10 now here's where we can be smart I can just swap these two around to find my equation for T I don't need to rearrange the whole thing again which I have seen students do we know that t is equal to Ln 12 - h / 10 all over K because again I could times both sides by T divide both sides by K that gives me Ln and what was the value when H is equal to 10 so 12 - 10 / 10 over - 0.09 98 let's sub this all in so we have Ln 12 - 10 I could have just said that's 1.2 but either way it's fine- 0.09 whoops 9 98 yep 16.1 Years cuz remember we want that unit and that right there is 10 marks now in my opinion the actual maths once we get going is not the difficult bit and that is a hopefully you've seen as a common theme with these videos whenever I point out these AAR videos the maths itself is not the issue I'm assuming if you're watching this series rearranging exponentials and logarithms is not the hardest thing in the world in fact this isn't even the hardest rearranging I've seen there's a harder rearranging question I could have shown you but it wouldn't be as hard as this question why because this relies on application and with gcse's most of you learn by learning methods and memorizing it and saying oh well if this is the case then I use this if this is the case use this very machine like this is where mats becomes a bit more creative you have to use your problem solving skills okay because again this is very different to what they've given you before the math skills are all the same but the beginning bit is the tricky part and if you look at it like I mean the beginning bits are are jokes right they're not even easy these first two steps which should gives you four out of the 10 marks by the way is a joke right 12al a BAL 10 like it's very quick but again it's not about the maths being hard like the numbers it's about the actual con concept and the application which is the tricky bit a lot of students got this question wrong this is a very very hard question all things considered so again when you're going through these questions highlight the information that they give you and then think about what makes sense to go first don't just jump straight in take a second just like I did I said well if I use this there's two unknowns here is three unknowns but this only have one unknown you should also always use the fact that they have to give you enough information to get the answer right so if there's two unknowns here three unknowns here it means this last piece of information has to be the way to get the first unknown because otherwise the question's impossible so keep that in mind and I'll definitely see you in the next question and