 Here we go mark your own. I didn't leave you much room to show work I'm gonna kind of have to do it all over the place here or you could have done on a scrap piece of paper on your test I'll make sure I show you leave you room to do work if log base x of three equals a and log base x of 25 equals b determine an expression for log base x of nine over five first thing I said is I'm not using the base change law Because it's base x base x base x so that's nice. I already removed one complete strategy Then I thought about this and Amanda. I said, you know what that nine? I bet goes with that three and That five I bet goes with that 25. I bet you that's why they picked those numbers. I Said one more thing Katie. I said well Mr. Dukes says if they give you a bunch of logs try and combine them if they give you one log try and break It up Shannon. They gave me one log. I'm gonna try and break it up. So I wrote it like this first of all log base x of nine minus log base x of five Why minus because the by the new side looks in the subtracting outside now what well What's right here it's all What's right here? I disagree. What's right here? You know what I really see here another three I see this is That not a three and is that not also a nine so I haven't violated any math rules It's still the same question. I've just made it look different. But why is that so handy Steph? What can I do with this here to here? Darn right. I think on my next line. I would write this as two log base x of three and I would drop the minus sign down Nathaniel, let's play that same game Nathaniel. What's right here? What's right here? I disagree What's right here? You know what here? This is also a 25. How could I write this as 25 to some power? 20 to not five squared not I don't want to write it as five as a 20 I want to write it as 25 to something ah 25 square root or 25 to the one half. I Think I can rewrite this as is that not still a five? Yes, but doesn't that look a little closer to that? Yes, and what can I do Stephanie with that one half? Also move it to the front Why is that so nice well as it turns out? I'm nearly done because this question says you know what log base x of three can be replaced with a And you know what log base x of 25 can be replaced with? B So I think this whole thing is really to a minus a half B which is oh Hey, how many you got that? Okay, so more of you getting that like I said there's kind of a science and an art to these and suddenly you see it and I Number two says what's an expression for the log of n if n equals that? Oh, you know what I'm gonna take the log of both sides because that will give me log of n on one side I Think the first thing I would do then is this The log of big n equals the log of a over 5b I Would glance and see if I saw that answer. No, I don't quite I See lots of answers with multiple logs. I think they want me to break this up So I will on top means what positive on the bottom means what? Just when we reversed it we said positives on top negatives on the bottom Reversing it is a bit is on top. It'll be positive when you pull it out of the log If it is on the bottom, it'll be negative when you pull out it. So it's gonna be log a minus log five minus log b and Oh, okay Now a lot of people make The mistake of picking D because they say oh those two are being multiplied and so multiplying means that no no no No, where is that B? Is it on the top or on the bottom? That B is on the bottom It's being multiplied on to everything on the bottom which means Fundamentally it's still dividing because it's on the bottom. It's negative number three. I thought was the toughest multiple choice I looked at this and I went No idea Haven't got a clue Well, no, I have a little bit of a clue Amanda, what's my base right here? What's my base here? Here Here here, you know what? I'm gonna base change and I'm gonna base change either to base B or base X Which one well my first thought was why don't I go base B because maybe something nice will happen with a 1 over B Changing into a base B. I have no idea where I'm going, but that was my reasoning So how would I write this whole thing is base B? I said, okay First I drew little lines. I wouldn't get confused in my work I said this is the same as the log base B of X all over the log base B of 1 over B Sorry for writing so small. I hope you can read that That's the base change law. Oh I said I well I looked at the denominator The log base B of 1 over B that I can evaluate What is the log base B of 1 of B to what power? Equals 1 over B No, I think I heard somebody say negative I think negative 1 so I said oh This is really just The log base B of X over a negative 1 And then I said Katie, you know dividing by negative one that just makes your whole answer negative and then I smiled I Said hey There it is there it is Just curious how many of you got all three of the multiple choice, right? So I mean I didn't water these ones down So if you did well played Number four, this is a log equation. We set our strategy for a logarithmic equation It was we want to write it as one thing equals one thing one term equals one term The right-hand side already is one term, so I'm good with that The left-hand side I'm adding two logs why that's the same as multiplying them if the bases are the same and they are Can I do this? No, no the logs don't cancel the logs would cancel it so if I had one log equals one log And even then they weren't technically canceling out a technique any longer most whatever. Oh Ooh, what did we do here? We got to this line and we had a log equals a number What did we do here? Amanda you're right. What? Yeah, this and the phrase the trigger phrase I tried to give you guys is if I know one I know both if I know the log equation I know the exponent equation what to the power of what equals what I'm pretty sure 12 to the power of one equals what's inside the log and I'm already sensing I'm probably gonna need more room. So mr. Dewey Move that up a little bit like that and I'm gonna hook and I'm gonna get rid of the one there What is 12 to the one? 12 I Ended up on my next line with 12 equals and I foiled this out I got a 6 you know what because this is with the x at the very end normally I try and do all this in my head But they got the x is negative and on the end. I'll be a bit more paranoid. I'll show all my steps So it's gonna be 6 minus 3 x minus 2 x plus x squared 12 equals I'll have an x squared I'll have a minus 5 x and I'll have a plus 6 What kind of an equation is this why it's a quadratic? How do I know? What's first thing I do before I even factor? What's the first thing I do? I make it equal to 0 0 equals x squared minus 5 x minus 6 because I would subtract 12 from Bossa Then I would try and factor this numbers that multiply to negative 6 and add to negative 5 I think negative 6 and Positive 1 I think my roots are 6 and negative 1 No We have to find out if Ryan does get his date after all If I put a 6 in here What's 3 take away 6? What sorry what oh? No, oh, no Let's try a little negative 1 right here. What's 3 minus minus 1? Well, I like that's okay. What's 2 minus minus 1? Ah, that's okay as well. So for 3 marks if you ended up with Most of this and a 6 and a negative 1 and you crossed out the 6 Otherwise, how do I give up part marks? That's a great question. Mr. Do it. How did I give up part marks on this? This is quiz 4 Semester 1 or semester 2? Doesn't say This semester 2 I hope I gave up part marks on this on my original answer key Let's find out Yeah, I knew I did so I would probably give you a half mark if you combine the logs a Half mark for realizing that 12 to the 1 equals what's inside the log a Half mark if I saw the lovely quadratic equation appear a half mark for the factors a half mark for the roots each of the roots But you would lose a half mark if you didn't cross out the extraneous Example 5 We call this an exponential equation How do I know the X is an exponent? How do I solve this take the log of both sides and no shortcuts here? Mr. Do it says too easy to make sloppy mistakes Move the exponents to the front with brackets get rid of the brackets by multiplying How many x terms do I have to are they on the same side? No Get them to the same side now look up Here is Kirsten my preference. This is not a math rule I find I make dumb mistakes with negatives Kirsten So even though the easiest way to get all the logs to the same side would be to minus this here I actually chose to plus this guy to this side because I got rid of that negative and Plus this guy to this side because I got rid of that negative Does that make sense? I liked it better So I wrote this like 2x log 3 plus X log 5 equals Log 3 you don't have to you could have got your X's to the same side. I Just Katie make dumb mistakes with negatives and I've learned that over the years How many x terms do I have to how many what I prefer Steph? Be wonderful if there's some kind of a grade 9 mathematical operation that I could you know pull out of my Back pocket as it were it I can factor an X out now what? Get the X by itself How it's so what's happening between the X or the bracket? So how would I move that ugly bracket over yep? It's math 8. It's ugly math 8 I agree, but it really on this level is math 8 X is gonna be the log of 3 all over 2 log 3 plus log 5 or you might have a negative here a negative here and a negative here It's the same answer. Oh Two decimal places mr. Dook. Okay pull out my trusty calculator. How many terms on top Carson? So I can probably be sloppy with brackets on the top How many terms on the bottom better open bracket? write the bottom close off the bottom Oh, and I was closing off each log bracket along the way without even thinking about it because I have to anyways The answer is point two nine. Yes zero point two nine if You didn't round off to two decimal places if you went point two eight eight I won't take a half mark off if you did round off to two decimal places And you said the answer is point two eight I'm taking a half mark off because rounding is math they learned around off properly point two eight eight It's point two nine. Now if you got that and you showed work you get two out of two Otherwise, how did I give up part marks? One two three four five six seven lines. Oh, what would be nice if there was eight lines? I Gave a half mark off a half mark for that line Half mark if you move the exponent to the front a half mark if you're able to get the x by itself and a half mark for The answer I'll be honest on a test. I probably make this worth three marks or maybe even four Which means if you get it it's free marks Which means if you can't get it it's throwing away a lot of work Some even clever enough to turn the page. Hey turn the page This is as tough as an exponential equation. We'll get It's still an exponential Nicole x is an exponent still gonna start out the same way take the log of both sides I am not gonna move the exponent to the front not yet Because if I move this exponent to the front I'd be saying it was on the five and it's not instead I'm gonna recognize it. So what's happening between the five and the bracket? That's the same as adding outside the log. I'm gonna break this log up Going it's gonna be the log Oh, and the two and the two x plus one those are now separate logs, mr. Do it and now I can move the exponents to the front the log five drops down like a domino I Get a two x plus one log two equals Three minus x Log seven who bought solution manuals So just a reminder when you're looking at the solution manuals What the author does now is he changes this to a decimal like to eight decimal places and this as well And he multiplies them in he changes that to a decimal I think as well and he just gathers like to like he treats it like a great eight math equation with decimals and things like That which works just fine except it doesn't work very well. If there's a non-calc section Or if they say for as an exact value So I still like to keep going and say, okay boom boom boom boom This is gonna be the log of five plus two x log two plus log two equals three log seven minus x log seven Get all my x's to the same side. I Guess I'll minus this to this side and plus and minus that to that side. I'll get log five plus log two minus three log seven equals negative x log seven minus two x log seven Tyson how many x terms do I have? To how many would I prefer? Be wonderful as a grade nine mathematical operable. Yeah, I can factor out the x now. I don't have to go over here log five plus log two minus three log seven equals need more room. Oh Problem, mr. Duk. I'm gonna have to go over here Yeah, that is so handy. I have to admit that is just so handy Factor out an x and I'll have a negative a log seven and a minus and a two Are they both log sevens or did I do a glitchy thing here? This? This is a log two isn't it and suddenly it became a log seven Did I say that there was easy to make dumbest strikes along the way if you're not really really careful? I'm telling you it is at least I caught it so that should be a log two did I fix it here? It's still a log two Okay, that all looks good So if I recovered this is going to be factoring out a two log two, I hope Holy smokes is the answer. Please tell me log five plus log two minus three log seven all over negative log seven minus two Log two is that correct boys and girls? People are nodding Of course I can always check because I'm going to type this monstrosity into my calculator and then before I hand my test in I'll graph left side graph right side and find where they cross and I better get the same answer Anyways, let's type this puppy Bracket Log five plus. Oh, no close off bracket. Mr. Duk plus Log two close bracket minus three log seven close bracket close bracket divided by bracket negative log seven close bracket minus two log two close bracket close bracket. Do you guys get 1.06? What does it say how many double play it to? 1.06 How would I give part marks out there for four marks one two three four five six seven eight lines I probably go a half mark for each line. I bet you let's see if that's what I did Yeah, look at that a half mark for each line And I also apparently added more space on my answer key so that I could fit this stuff in Okay What would you do mr. Do it if you did that dumbest rake like made a two into a seven overall I'd probably work yours out because you've accidentally changed the equation We call that a transcription error in the marking realm and I'd probably give you three and a half out of four Number seven another log equation Mr.. Duk says our strategy is to write this as one thing equals one thing well the right-hand side isn't too hideous This is going to be the log of two times 22 minus 2x I Can't cancel the logs out. I need to move this to up to there. I need to go log Three minus x all squared now. Do I have one log equals one log? Okay, then the logs can't okay. I'm taking the analog about find the logs cancel and I'll end up with an equation that looks like this Three minus x squared Equals to bracket 22 minus 2x. I'll get rid of brackets 44 minus 4x This is 3 minus x times 3 minus x I think it ends up being 9 minus 6x plus x squared when you foil it out when you write the bracket twice and do the foil Shannon, what kind of an equation is this? How do I know it's got a squared? What am I going to do first before I do anything else? Gonna make it equal to zero. I think I'm gonna minus and plus these guys over I'm gonna get x squared. I'll plus 4x to both sides. That gives me a minus 2x I'll minus 44 from both sides 9 take away 44 is you get minus 35 I'll factor are their numbers that multiply to negative 35 and add to negative 2 if I didn't factor I'd quadratic formula Oh, no, there is it does factor x minus 7 x plus 5 Equals zero. What are my roots 7 and negative 5? Oh, no, we got to see if Ryan gets a date after all Go back to the original question if I put a 7 right there. What's 3 take away 7? Negative 4. Oh, no, Ryan. I'm so sorry Let's try the negative 5 3 minus minus 5. I'm good with that 22 minus negative. I'm good with that too. That works. Okay, and here there is no x So I ignored that term there. So this one looks like it's okay terms of part marks I Did that half mark for combining the logs a half mark for getting rid of the logs a half mark for doing some algebra Half mark for the quadratic half mark for the factors Half mark for each solution, but a half mark off if you didn't cross out the extraneous route. Oh Hey, hey, hey What if I'd also ended up crossing this one out? What if that one hadn't worked? How many solutions are there then? Yes, you may have no solution. It's possible What that's really saying Carson is this graph never crosses this graph it could have It'll be a fluke. It doesn't happen very often, but it's out there if you could give yourself a lovely score Please out of 16. Is it out of 16 and those you the world may can you awake? Can you put a great big omit on your quiz? So I don't give you a zero Can you open your books to last day? We looked at logarithmic scales. We looked at Richter scale Decibels and pH And we said that they were all actually scales based on powers of 10 Oh, no decibels were based on powers of 10 divided by 10 because and I remembered that because of the desa Prefix I could remember that we said as it turns out when you're looking at how sound works When you're looking at how earthquakes work when you're looking at how pH works the diff a difference of one Actually means ten times stronger. So an earthquake of Richter scale three and an earthquake of Richter scale five It's not five take away two twice as strong It's five take away to ten to the two a hundred times as strong. I gave you some questions to try and I'll start out by saying any questions from the homework now is your chance To ask and I'll probably talking about the very last question number 12. I believe it was in a second So relax, but any of these all went good What was the whole thing and this you are gonna have to memorize it's I said It's always gonna be 10 to the first minus second equals how many times stronger That you'll have to That if you're making a steady sheet up or I think we wrote it down a few times in our notes Like that's what I would kind of star or put a little asterisk next to or or or something In fact, I even had you guys repeat it. I think oh, please I wrote it in the notes somewhere. Did I not? Please tell me I wrote it down in the notes somewhere. Oh there at least once there There's what I fall back on so hey, let's look at number 12. That's right number. Yeah, no number 11 This this is the one all right There is an easy way to a clever way to do number 11 showing no work Because it's multiple choice and Then there's the mathematical way. I'm gonna do the easy way first says this Two different jets are flying together in an air show anybody ever been to the arbitrary air show Okay, if you haven't well worth going at least once it's quite a shame Each with a sound level of 120 decimals Then the approximate total so if there's two jets first of all Steph if one jet is 120 Will two jets possibly be less That's a dumb answer. Will two jets possibly be less That's a dumb answer Maybe Oh at air shows Do people regularly have their fillings shatter and screaming pain because they're exposed to 240 decibels of sound When we said the painful level was 160 decibels Do people go to air shows and do we have ambulances coming out of air shows time after time after time Andrew what's the obvious answer by thinking it through that way Now the real question is all right duic Prove it Okay, okay Here's what we're really saying there's one plane Remember we said decibels you divide by 10. There's one plane. Yes Can I instead of writing 120 over 10? Can I put a 12 there now? Here's two planes Yes that equals 10 to what power And once I know the exponent, I should be able to figure out what decibel decibel level that is Katie is that okay? Now here is where I would cheat just a little bit Here's where I would cheat just a little bit. What's my base right here What's my base right here? What's my base right here? I'm actually going to bring them to the same side because this is an this is an exponential equation But it does have part of the same base which we don't often deal with I'm going to divide this to this side I'm going to write this as 2 equals 10 to the x divided by 10 to the 12th So far so good. I have no idea where I'm going, but that looks nicer Oh wait, wait, wait, wait, are my bases the same when you're dividing with exponents? What did you do when your bases were the same? There's another way I can write that right hand side Yet minusing I connect you know what the equation is actually this When I do some fancy schmancy algebra, Nicole Where's the x sitting, Nicole? I know I did that on purpose mid-yon that was fun. Where's the x sitting? So I'm going to log both sides base 10 And Nicole because I'm in a rush I'm actually going to take a shortcut and move the exponents down to the front All on the same line I would never do that on a tester or quiz because I've told you well you saw I made a dumb mistake today in my answer key um This is interesting. What is the log of 10? Sorry. Oh, that's even nice. So if I hear you correctly, I get this log 2 equals x minus 12 How would I get the x by itself? Please Thank you plus 12. Okay. I get this x is going to be the log of 2 Plus 12 it's like we're running off to my calculator Log 2 Plus 12 and I get that x is 12.3 Hey, mr. Dewick wait a minute. That was answer. Hey, we're not done. We're not done. We're not done Okay By the way, I'm not going to give you one quite this tough I'm only doing this for the nerdiness of it and it is good practice to find the other ones easier Here's what we're saying then two jets Is the same as that but ryan That's in bells Do you remember for decibels? We actually divided the exponent by 10 if you want the answer in decibels It's 123 decibels divided by 10 that's how we get 123 decibels It we had to reverse the procedure remember Amanda I had 120 over 10 up here and then I said I'm gonna write that as a 10 So I went backwards to say well instead of 120 what would be up there 123 So you can use that second method. I would use my first clever method if I were you Yes Did anybody do it that way but anybody noticed that it has to be see Yeah You are smarter than your brother. I mean good Yes, I'll upload this video and you can drag your brother in front of this video and fast forward right to here and say Look, look, look, mr. Dewick is still making fun of you still. What did you do to him? Is that all right? So how will this show up on your test? You know what? I'm probably going to put one multiple choice Logarithmic scale question and one written logarithmic scale question and it'll be on ph decibels or rickterscale Okay You know what the funny thing is I've done that. What now five or six times. I can still see a few eyes What's he yelling for that's stupid Okay It's only so much I can do turn if you would please To page 203 we skipped one more lesson because we're going to do that one actually Actowards page 203 page 203 Under the b2 Under the i o under the n 3 page 203 Page 15 squared minus 22 So we just looked at applications of log scales now. We're going to look at applications of exponential growth and I have to start out by Scaring you is the wrong. I got to give you a little shock therapy This is probably on my list one of the top five toughest things we'll do all year Because what we're really going to be looking at here katie is word problems now. Don't freak out Don't freak out. I'm going to teach you how to dissect them We're going to be doing word problems and we're going to end up with exponential equations So we're going to have to use logs to solve them. It's a tough concept. It is nerdily cool Trust me at least once in here. Oh, that's how they figure some of that stuff out So read along with me. It says this we're probably going to go right to almost the tone today I'm going to probably do part one of the lesson today I sign a couple of the questions for homework and then we're really going to jump into things the next day Remember an exponential function is a function whose equation looks like that There is a base the x is an exponent and it's there may or may not be a coefficient Okay, this time there is almost always going to be a coefficient because now we're looking at real world Problems and the numbers are never as nice as the fake ones that we can try for graphing Your base is rarely going to be a tool. It's usually going to be a decimal Oh, and we said if as long as a is positive If your base is bigger than one, it's an exponential growth Kirsten looks like that if your base is less than one x an exponential decay. It's getting smaller says this Blah blah blah blah blah I'm not going to look at that paragraph yet because that paragraph refers to the previous lesson Which we've skipped right now Instead I'm going to tell you that every single exponential growth equation looks as follows Can you turn the page please? And go to page 205 at the bottom of page 205 This right here is the template That we're going to memorize we're going to say every single exponential growth equation Looks like this Although I'll be honest rag jeep. I get a little sloppy. I don't put the t subscript there because that's saying a of t I don't want to complicate it with the function notation What I write whenever they give me an exponential growth word problem And you'll learn how to recognize them pretty easily the first thing that I do is I say a Equals a zero c to the t over p. You don't need to write that right now Yes, you do. Can you turn back please to the beginning of the lesson? And write where that paragraph of blah blah blah blah blah is Where it says writing an equation using y equals a b to the x we're going to say actually it's going to be a equals a zero c to the t over p Say what write that down Put a box or a star or something you do need to memorize this equation Now you'll probably write it out so often if you discipline yourself jessica to write it down Every time you're doing a question even if you've just written it in the question above That's probably the easiest way that you bring will just memorize it because you're naturally lazy Let's talk about what all of these mean the letter capital a means the final Amount that's why we use a for amount So whatever number you get here. That's what you end up with It's what you finished with They sometimes put a t the final amount with respect to time. I don't like putting that little t subscript That's too many. I got confusing ashley. So I won't make you even though they did in the book physics people What do you think a zero stands for i'll give you a hint. It's a certain type of an amount What did the zero stand for often in physics? Okay, this is your initial amount That's at least nice and consistent with physics notation C is your growth Rate it's also called your growth constant Which is why they use the letter c for constant I always use the letter r for rate, but I want to be consistent with your book. So I'll use theirs t physics people. What does lowercase t always stand for? time I'm going to add a word in front of it Total time And the lowercase p stands for the growth period The population doubles every five years your growth period would be five Population doubles every six months your growth period would be point five half a year Population doubles every year your growth rate would be your growth period would be one Okay Now if you memorize this then katie it kind of becomes fairly plug and chuggish says this Example one in 2002 the university population of a country was a hundred and sixty thousand Increasing at an annual rate of four point five percent A says write in equations represent the university population as a function of time Now i'm going to cross out the letter p and i'm going to put a for final amount just this once And instead of n years i'm going to put t time Okay And we would start out by saying final amount equals initial amount Times c to the power of t over p and now we would plug stuff in This 2002 we're going to come back to What's this hundred and sixty thousand? Your initial amount if you can put an equal sign and in front of the of the where c there's going to be there's your initial amount 160 000 katie in this case the final amount is what they're asking us to find trust me. I know there's going to be an a there Final amount equals but your initial amount is 160 000 What's your growth rate? It's often a percentage. Can you see it? Okay, don't write four point five percent yet. First of all four point five what? Okay, we can never do math with a percent. We always have to change it to a decimal So what's four point five percent as a decimal? math eight What's fifty don't write this down. What's fifty percent as a decimal? point five What's two percent as a decimal? What's five percent as a decimal point zero five, so what's four point five percent as a decimal? Put a bracket here leave a small space, but put a point zero four five, but leave a space in front of the point Because this isn't quite right It's actually that as your growth rate one point zero four five Why is it that? look up First of all, what's one as a percentage? No, what's one as a percentage? Carson, thank you for remembering your math eight. Here's what you're saying Your original population plus four and a half percent every year That's why you have to have a one in front of it katie because your original population is still there And it's increasing by four and a half percent Your growth rate is four point zero five to the power of t over What's the growth period? It says the population is increasing at an annual rate. That means it's increasing every year Every one year your growth period here would be one Trust me. It'll get easier This is your equation your final amount is equal to your initial amount times your percentage increase plus one To the power of time divided by however long the growth period is annual means every year What if it said semi-annually every half year it'd be a point five here What if it said by annually it'd be every two years there'd be a two here What if it said every decade It'd be a 10 Okay B says determine the university population in the year 2005. What do they want me to determine? What do they want me to determine Shannon? Read Which of these is the final university population? This This You wrote this down didn't you? You write that down So what does the a stand for look it down? Look at your notes. Look at your notes. Look at your notes. Look at your notes Let's keep looking at your notes See that right? Okay. Keep looking there. Look down What is what did you write next to the a? Wouldn't that be the university population? They want us to find a Okay, in other words if they want us to find the university population in the year 2005 The university population is going to be 160 000 times 1.045 To the power of what? What year are we in? What year are we in? No, no, what year are we in according to this question right here right now? What year did this question start? What am I going to put right here? Here's my variable. Is my variable exponent? No, this is not logs. This is go to your calculator type that in plug and chug. What is 1600 times 1.045 cubed? Calculators please 160 000 times 1.045 cubed What do you get? 4.67 we're starting with 160 000 people. There's no way we have 4.67 I don't think you've multiplied by a hundred and six 160 000 times 1.045 cubed Right What do you get? You got a hundred and eighty two thousand five hundred and eight six point six Yes hundred and eight two thousand five hundred and eighty Can we have point six of a person? Well, there's midsuit. No, can we have point six of a person? Let's round up hundred and eighty two five eight Seven Yes Yes Okay C C says if the population continues to grow at this rate Determine the number of years to the nearest year for the population to double First thing I'm going to do is I'm going to write down a equals a zero C to the t over p First thing I'm going to do is I'm going to write down a equals a zero C to the t over p Partly because it helps me to memorize the thing and partly because that's how I'm going to approach solving this kassandra. Thank you First thing we're all going to do is write down a equals a zero c to the t over p And then we're going to do in physics what I call deficking renalist our data What was a zero? What was my initial population in the information that they gave me? I'm going to write over here. Do I know the final population? I do indirectly What's the final population supposed to be according to this particular question? How'd you get 320,000? More specific. What was the word that told you double? What was my growth constant my growth rate? We were increasing by what percent? 4.5 percent, which means 1.045 And we were increasing every two years every one year every six months every one month every week. What were we increasing? It said increasing annually, which means increasing every one year You guys okay because I'm seeing lots of giggling back there. I hope it's math related What are they asking me to find in this question out of all of these? What's missing? What don't I know? What are they asking me to find? Nicole Did you say I thought you said sorry? What did you say t? Let's plug everything else in final amount 320,000 Equals initial amount 160,000 times 1.045 To the t over one am I going to write the over one though? Where is the variable sitting now amanda? Oh, how will I solve this then? This is why it's part of this unit. We're going to use logs, but first We're going to do something clever before we take the log of both sides because I don't have an exponent over here I'm going to get this exponent by itself. I'm going to move this 160,000 over how what's this 160,000 doing to the bracket mathematically? I'm going to divide I'm going to divide by 160,000 Divide by 160,000 And I don't need to reach for my calculator andrew. Are you serious because I know that 320,000 divided by 160,000 andrew. I'm positive is to No calculator required How will I solve this? How will I solve this? log both sides Okay, I was bad. I also moved the exponent to the front on the same line This is about the only time I let myself take a shortcut because these are usually pretty clean when you get to this line Holly, how would I get the t by itself? More specific divide by what? I heard you say divide by ah, yes If I'm not mistaken the amount of time is going to be the log of two divided by The log of 1.045 How many years after 2002 will this university or will this pop yeah university population Of this country have doubled sorry It says to the nearest year okay, um Would that be helpful to know if you were a country would that influence how many universities you were building over those years and at what rate? Yeah Because if university population is doubling in 16 years you better make sure in 16 years you got twice as many university spots Okay, this is exponential growth Turn the page figured out good example two The number of fish in a lake is decreasing. I want you to all to underline the word decreasing By five percent each year as a result of overfishing Says write an equation to represent the number of fish after t years use n zero to represent the initial population And n of t to represent the final population. Okay At the top of the page here. I'm going to write a equals a zero c to the t over p That's my generic equation my template that I memorized Except instead of a zero. What do they want me to use for the initial population? Meet the question It's a Instead of a zero. What do they want me to use for the initial population? Instead of a zero Thank you They want me to use n of t for my final population. Okay. I'll use their variable and make them happy What percent are we decreasing by? What percent are we decreasing by please read the question If we're losing five percent, what percent is left behind? 95 In other words for percentages. It's one plus if you're gaining One plus four point five percent if you're losing it's one minus If you're losing one minus five percent to the power of t and It says each year each year means the period is just one So it's key over one. This is your equation Right now, of course, it has one two three variables. We can't solve this But I bet you in b they're going to give us some more information that says this If there were 2,500 fish present in january 2000, how many would you expect to be present in january 2005? What's this 2,500? I think it's the initial population What do they want me to find in this question? What do they want me to find in this question? How many oh, you know what final population? That must mean somehow they've given me the time. How many years is this population talking about this question talking about? Oh, how'd you get the five? Oh 2000 2005, okay five Is my variable sitting as an exponent now, uh Cassandra is my variable already by itself Oh, then go to your calculators, please What is 2,500 times 0.95 to the fifth? What population is left? 1,900 and Uh, can we have 0.5 of a fish? What can we do on minnow? That's probably about 1,930 to be real honest see After how many years to the nearest tenth would it take for the population to reduce to half the number? In january 2000 Now this is where the fact that people don't understand how exponential growth and decay Really hammers us in politics or in society We're going to solve this in a second But a lot of people someone without a math background would say well, you're losing five percent How long to get to 50 percent uh five percent? 10 years they'll say oh, we got 10 years before we only got half left And tell you the answer is far less than 10 years Because this part of the curve is Decaying very very very very steeply. Let's plug in numbers What's my initial amount in january 2000? 2,500 What's my final amount that they want this to have? It says reduced to half. So how many fish do we want left? 1250 0.95 and they want how long would it take how many years t is my variable Where is t sitting? Exponent How will I solve it? Well, I'll eventually take the log of both sides, but is there a coefficient in front? Is there an exponent over here that I'm actually going to go divide by 2,500 Divide by 2,500 and Andrew without a calculator. Can you please tell me what 1250 divided by 2,500 will be? This divided by that what will it be? You know how I knew it would be a half? Here's my equation How will I solve this one? Log both sides I'll get the log of 0.5 equals and I'm going to be bad and move the exponent to the front log of 0.95 Christian, how would I get the t by itself? What's happening between the t and the log? So how will I move the log over? Are you saying divide by log? No, I think you're whispering it or maybe even thinking it because I'm having a read. Is that what you said? A little more volume next time? No idea what that was. I saw this I think you said divide by log. Yeah, I agree T is going to be cursed in that log of 0.5 divided by the log of 0.95 I'm going to tell you You don't have 10 years until you're down 50 percent. How many years? sorry 14 13.5 says to the nearest what in the question? Can't read the instructions for rounding right? You got 13 and a half years You okay? good We're going to come back to continuous growth and decay in a second What I'm going to assign you for homework right now is questions where they're giving you the equation We have been deriving the equation which is the tricky part We're going to practice that way more when I see you guys next but for now If you turn to page 207 you can do number one number two And I think For now Think Yep, that's going to be it Okay