 Hi, whoever's watching this online. Sorry, I forgot to record if you want to get the front page of the quiz cheat off a friend Mark it yourself. Number eight That's okay. This is the really important question. This is a tough question. This is the holy, holy question anyways I looked at this and I went, huh? Well, no, no I said I'm gonna be stubborn and clever and I got to try something. I Reasoned my way to this Nicole. I said there's a seven a seven a nine and a three I said I think these two go together And I said there's a two a two a nine a three. I said I think these two go together I said I'm gonna pick up this first log and I'm gonna try and turn it into that which definitely means I'm pulling up my base change law I want to rewrite it as base three. I don't want to change the seven. I'm good with the seven So if I rewrite it as base three, I'll get this the log base three of seven over the log base three of nine That's the base change law. I Said oh, what is the log base three of nine? That's kind of nice. Oh According to this question, what is the log base three of seven? What can I replace it with? I? Said you know what this guy is actually x over two I'll come back to that I Picked up this guy. I wrote it down I said I you know what I'm good with the two but instead of a three What would I like sitting inside the log instead? What would I prefer sitting there instead of a three? But it is a three How can I write a three is enough? Ah, I said is nine to the one half still three yep But what can I do with one half square root, right? What can I do with that one half exponent? What can I do with that one half exponent darn right? I can move it to the front. Absolutely. I'll get this a One half in front and then a log base two of nine Which is one half oh? According to this question. What can I replace the log base two of nine with? Because I asked you to what can I replace the law? You know what it's been 15 years, and I still I'm not tired of that joke. I will get tired of it all year long yet. Why in fact I think This is the same as a half y So this plus this is the same as or you could write it as Just times in by a half is the same as dividing by two and that looks a little prettier Or since they're both over two and I have a common denominator Ryan you could also have written it like this Because it turns out there's a common denominator so I can combine the fractions I would give you probably almost certainly one mark for the getting the x term and one mark for getting the y term And there's your two marks That's a tricky one Although it won't be in a few days Just curious how many got that only a couple Number nine it says right as a single logarithm What's my base in number nine? What's the base in number nine ten? I had some students in my last class. They said they thought it was a I said Well, first of all if it was an a the a would be really small next to the g but also it would be to log base a of There's got to be something in the law the a's got to be inside the log First thing I did was I moved this up to there and I moved this donk onto there And I said this is really the log of a squared plus the log of a to the fifth cubed Screen froze. Oh, what's going on today? Working good. Okay, just picked it up again should be a beep right about now Then there should be about ten nine eight seven six Or three we're back Thank you The exponent would go on the a to the fifth By the way, it's not going to end up being an a to the eighth What's it going to end up being? Yeah, this is really The log of a squared plus the luck of a to the 15th Which is the log of a squared times a to the 15th This whole thing simplifies to the log of a to the 17th power or you could move the 17 to the front In fact, there's another way to get this same answer. You could have done this Or you could have gone to log a plus 15 log a you could have moved the five to the front. How's that help mr. Duk? Well What's 2x plus 15x 17x what's to log a plus 15 log a pretty sure it's 17 log a so there's two ways to get there both are valid Both are fine Number 10 graph this we said our strategy is going to be Figure out what it's based on write the key points and then do the transformations So I said to myself self. This is based on y equals one half to the x Actually, I said this Steph. I said I'm going to graph the points for two to the x negative 1.5 zero one one two two four three eight Asymptote y equals zero Now that would be the graph For this I want the graph for that I guess that would make this positive this negative this negative this negative and oh that what's a why it stays the same This graph is based on those points Katie Now I'm going to do the transformations Plus two to left minus four four down Two left instead of positive one negative one negative two negative three negative four Negative five. That's a why so that doesn't move at all four down Negative three point five negative three negative two zero four four down Negative four this graph is going to have an asymptote at negative four This graph is going to go through negative one three point negative three point five Negative two negative three Negative three negative two negative four zero Negative five one of those three four Kirsten this graph is going to look Like that How would I give marks out I would give one mark out for the asymptote I? Would give one mark out if you had a shape like that if you included It was an exponential graph, and then you get two marks for the points now I said at least three you didn't have to do all five like I did But if you did at least three take a half mark off for any point That's incorrect in the wrong place out of two. That's how I would mark this one That makes sense And if you could give yourself a score out of count a mate team yo so There's quiz for We looked last day at identities. I think the homework was around page 153 154 sorry And what I said to you was I'm not going to ask you or give you an identity question on your test But this is great practice of log rules And you will have trig identities later on We also looked at if then questions last class So I'm going to start out by saying hey any questions you would like me to go over now is your chance to ask can a or B or both When I circle five and there's an A and B that probably means both anyways, maybe both I'll probably try five B on your own. We'll see Any others besides five? Yeah Okay, any others So usually I found kids get three For number three probably what I would have done is rewritten This is x to the negative one and then move the exponent to the front and you're done There's other ways to get there Roxanne first thing I would do is this there's my tea table. That's my quick easy way to do a tea table What's which side Roxanne am I going to work on? I think it's pretty obvious which is the uglier side Absolutely by the way look at the right side now. It says log base C of C What is the log base C of C? So I'm gonna leave it as log base C of C But I'm gonna keep in mind maybe what will happen on the left side is instead of getting log base C of C Maybe I'll reach the point where I get fractions where everything cancels, which is also a one That makes sense What's my base on the left hand side? There's two of them. What are my bases on the left hand side? I'm gonna go base change and I'm gonna try writing it as base C because that's what the right hand side So if I rewrote this as base C it would be the log base C of B over The log base C of A that was what the first log would look like as base C Is that okay so far Roxanne? And I would go the second one is gonna be the log base C of A Over the log base C of B. Is that okay as well? Does anyone yet see that we're done? Because you know what I have here Don't I have the same thing here and here same thing here and here? Same thing here and here same thing here and here how many x's on top one how many x's on the bottom one How many left and where? I'm gonna argue That I think That cancels Yes, and That cancels when you have a fraction and everything cancels. What does that simplify to not zero? What does it simplify to? Ah? Ah And you know what I'm gonna go over here, and I'll rewrite that as a one There's a left-hand side equal the right-hand side. Well, then number five number five Five a There is a less clever and an uber clever way to do number five a The less clever would be to say well. I'll write the left side is base B Or I'll write the right side as base a Andrew that would be the less clever way You want to do it the uber clever way, don't you I can tell so I'm gonna do this Stay on this page Andrew stay on this page, but I really want you to look closely at this stay on this page Stay on this page Do you agree with me that our strategy should be some type of a base change? Okay I'm gonna go zipping off to when we did the base change law The base change law looked like this When you had the same base top and bottom You could take these and write them like that. I don't know if you just saw it I saw your body language change. Maybe you did get your nerdy adrenaline rush. Let's see I'm gonna argue here Andrew that I can rewrite this here as the log base q of P because if I gave you that and said write it as base a that's what you would do. We're reversing the base change I'm gonna argue that on this side. I could write this as the log base q of P because if I gave you this and said write it as base B That's what you would get and oh I do have left side equals right side That's the uber clever way to realize I can reverse the base change to its simplest form and Although I could do it the long way. I'm gonna say I'd do it that way I'll let you try 5 B on your own Now I said to you I said to you Kirsten that Identities would not be on the test and they will not be t-table identities Something like numbers eight and nine are gonna be on your test. I'm gonna give you some log expressions to simplify Okay, to me. That's not my identity. That's just practicing your log rules So make sure you're good on eight and nine They are completely fair game We're gonna jump we're gonna skip a bunch of lessons. We're gonna jump all the way over to page 213 page 213 What we just did Jessica's we skipped a little four lesson mini unit. I always do that at the end of the year It's called series and sequences. I Do it at the end of the year because that's my snow day hedge bet Although it's four lessons and if I have four days, I'll take four days I can cram it into two if I need to In fact, I could do it in one god awful brutal one lesson if I really needed to Now that was when we had the mandatory provincial this year That's my if I had to skip one little tiny unit and cross up a couple of questions on the final exam That'll be the one although. It's nearly cool. That'll be the one What we're looking at for the next three or four days actually is some applications of Exponential and logarithmic functions wasn't a coincidence that began class with that little snow graph Where we looking at today or what are called logarithmic scales and the first application of logarithmic scales is in earthquakes Richter scale This is how Richter scale works Richter scale is named after the American seismologist Charles Richter And it's used to measure the magnitude of an earthquake. In fact, the magnitude of an earthquake is the measure of how much energy was released They get it from the logarithm of the amplitude of waves recorded by seismographs You don't need to know that all you need to know is how Richter scale works By the way, the 2011 Japan earthquake was a 9.0 on the Richter scale. Here's how Richter scale works look up If earth look up means look up if earthquake a has a Richter scale of four and Earthquake B has a Richter scale of two Earthquake a is not twice as strong. What's four take away two? What's four take away two? It's ten squared a hundred times as strong Richter scale is a ten fold scale if you have earthquake a that's five on the Richter scale And earthquake B is two on the Richter scale What's five take away two? Ten to the third one thousand times as strong Okay, what if you have an earthquake of Richter scale like in Japan nine and Oh about 12 years ago. Remember that little mini earthquake that we had here, which was a three Nine take away three is six ten to the sixth the earthquake in Japan was a million times as strong Which means a million times more damage Which is why it did so much damage It's why on the news when you hear oh an earthquake of Richter scale seven And you compare it to an earthquake of Richter scale six you might be saying well different to seven of the six Isn't that much? Why is there so much more damage? No, no, no seven take away six is one ten to the one ten times more damage This is how Richter scale works So example one says an earthquake of magnitude eight is how many times as intense as an earthquake of magnitude seven Ten to the one I normally don't write the one I will this time so you can see where the exponent came from an earthquake of magnitude seven is how many times as intense as an earthquake of magnitude three Ten to the fourth 10,000 times stronger I don't like the way C is worded an earthquake of magnitude four is how many times as intense as an earthquake of Now, it is going to be ten to the negative two, ten to the negative two, or one one hundredth, except I don't think that's as intense. I think that's weaker. I wish they would say is how many times weaker, one one hundredth, but whatever. A four point eight compared to a six point eight is also a ten to the negative two if we're looking at the four point eight as our reference point. One one hundredth times as strong. E. The nineteen seventy six earthquake in Italy was how many times as intense as the nineteen six? Okay. Italy. Seventy six in Italy was six point five, Columbia five point five, ten to the one, or ten. Nineteen thirty three earthquake in Japan, how many times as intense as the nineteen sixty six earthquake in Turkey? Thirty three Japan eight point nine, Turkey six point nine, ten square, hundred times stronger. Turn the page. They're deriving how you can compare earthquakes. I really think they over complicate how to get there. Here is Mr. Duick's simplified version. This is what we're going to put in a little box right here. Ten to the first earthquake minus the second earthquake equals how many times stronger or weaker. Ten to the first one they give you minus the second one they give you. Ten to the first minus second. Example two says this. How many times more intense was the nineteen eighty five Mexico earthquake? That's the first one. Ten to the power of the first one minus the second one equals how many times as intense. Ten, what is eight point one take away six point nine? You have to do the exponent first like that math, right? Eight point one take away six point nine, eight point one take away eight is, sorry, take away six is two point one take away point. One point two? No? You need your calculators out boys and girls. What is eight point one take away six point nine? Is it one point two? No answer. Okay, I'll wait. It means more homework for you guys. Is it one point two? Yes? Okay. Now, what is ten to the one point two? It's going to be a decimal. What'd you get? Ten to the power of one point two. Ten to the power of one point two is, one point two, not two, two, one point two is, oh, 15, oh, wait a minute, what's it say? Answer to the what? Okay, 16, 16 times stronger, roughly 16 times more damage. Why do I say roughly? It also depends on the construction. The one in the Dominican Republic and Haiti and Port-au-Prince a couple of years ago wasn't that big, but the construction there was so flimsy that it really damaged everything as well. The one in Japan was not only, because Japan builds everything earthquake truth. They take it seriously, but the one last year was so huge it still did damage. We're really more interested in example three. Example three says a major earthquake of seven point five is three hundred and seventy five times as intense as a minor earthquake. Find the magnitude of the minor earthquake. Ten to the first minus second equals eight times. How many times stronger does it say this first earthquake was than the second earthquake? Three hundred and seventy five? What was the magnitude of the first earthquake? The bigger one. Seven point five minus, what was the magnitude of the minor one? We don't know. X. Cassandra, where is the X sitting exponent? You know what I'm going to have to do to both sides? Take the log. This is why we do it in this unit. Now we can actually solve this. We're going to take the log of both sides. We're going to get the log of ten to the seven point five minus X equals the log of three hundred and seventy five. What can I do with the exponent now? Since it's a binomial exponent, when I move it to the front, what will I remember? Brackets. Yes. Yes. Yes. You would have remembered the brackets. Right, Holly? Not. Holly, what's my base when I don't write a base? Are you saying this is the log base ten of ten? What is the log base ten of ten? This is one times the bracket, which really means for all intents and purposes that log base ten of ten vanishes. My next line really looks like this. Seven point five minus X equals the log of three hundred and seventy five. Let's get the X by itself. Nicole, I think the easiest way to get the X by itself is going to be move the X to that side by piecing it and minus this to that side by minusing it. I think the easiest way to get the X by itself is going to be X equals seven point five take away the log of three hundred and seventy five. What is the Richter scale measurement of this smaller earthquake? It's going to be four point something or five point something, I think, if my math is correct. Four point nine? Yes? No? Yes? Yeah. Four point nine on the Richter scale. Just to give you a bit more perspective, a train going by, a big freight train when you can feel it if you're waiting in the car about three or four cars away, you can feel it through the ground a little bit. It's Richter scale about point eight. Semi-truck driving by is Richter scale about point three. The Japan earthquake last year was stunningly big. If you're a nerd at all, every so often Nova on PBS has been doing specials on the Japan earthquake looking at some of the numbers. There was a seaside town that had built a 30 foot high, 30 foot high sea wall because they had done the arithmetic and even if there was an earthquake of magnitude nine, the biggest tsunami you would get was 25 feet high. So they built a five foot safety margin. They still got flooded. The wave went over the wall and the engineers were going, wait a minute, we've done the math time and time again. The wave should not have been 30 feet high. It should have been 25 feet high. Turns out the earthquake was so strong, their town sank by 15 feet. The whole coastline dropped by 15 feet and there went their safety margin. They hadn't thought of the whole island sinking again, I mean really that was that strong. Go watch some of the specials on the Japan earthquake on Discovery Channel or PBS or whatever. It'll knock your socks off how strong that earthquake was. We sit on an earthquake fault too. Almost every one of our high schools is not up to earthquake code. This one is not. I think in 2000 the government estimated it would take about $10 billion to get all of the high schools up to code and they said, we're going to do it. I believe last I checked they've done two schools out of the province because they don't have $10 billion and they don't want to spend the money because they could spend it on the schools but they wouldn't be able to give everybody all the free stuff. They'd lose the next election for sure. Of course the obvious answer is if there is an earthquake and all the schools come down won't that cost far more than $10 billion to rebuild all the schools? I wish the government would think that way but no government, NDP or liberal or whoever, no government will think about spending money 15 years from now to save an issue 15 years from now. They don't work that way. Three words duck and cover. There's a reason we do the earthquake drills, right? So Richter scale, Katie is 10 to the first minus second equals how many times stronger? Katie what is it? 10 to the what? First minus second equals how many times stronger? Katie what is it? 10 to the what? How many times stronger? Cassandra what is it? 10 to the first minus second, I need you all with me, is how many times stronger? I'm doing this for a reason, for an important reason. Shannon what is Richter scale? 10 to the power of? Really guys? First minus second equals how many times stronger? All of us all together, 10 to the power of what? Equals one more time all of us, 10 to the power of the minus the second equals, because I'd like you to skip loudness of sound and I'd like you to turn please to page 217. PH. PH. Who's in camp 12? I think you'll look at this in more detail. This is Mr. Dewick's down and dirty simplified, hey here's how PH scale works. So it says this, PH scale. In 1909 Soren Sorenson, if you had a kid would he be Soren Sorenson son? Get a boy? No? Okay. Anyways a Dutch chemist introduced the term PH representing the expression the power of hydrogen to measure the extreme wide range of hydrogen ion concentration in substances. I have no idea what that sentence means, I don't care. The PH scale measures the range of hydrogen ion concentration by determining the acidity or the alkalinity of a solution. I have no idea what that means, I know acid means don't put it in your eye. I've seen enough movies to, okay acid bad, alkaline I think has something to do with the word base or something. I don't, I took chem 11, I don't care right now. I can read a chart. Says the scale measures from 0 to 14 with 7 representing, with values below 7 representing increasing acidity and values above 7 representing increasing alkalinity. Say what? Look at the chart. I think if you're to the right, sorry to the left, your acid, if you're to the right, your alkaline and it's just like Richter scale. Every difference of 1 is actually 10 to the power. In fact, one might say that PH is 10 to the power of first minus second equals how many times more acid or alkaline, which is why I had you repeat that little chant. Show you what I mean. Example 8. It says complete the value using the approximate PH values given. Tomato juice is how many times as acid as pure water? What's tomato juice? Four. What's pure water? Seven. 10 to the third and it's acid because four is to the left of seven and it says acid is that direction. Tomato juice is a thousand times more acid than pure water. Eggs are how many times as alkaline as pure water? Well, what's the PH of eggs? Eight. What's the PH of pure water? Seven. What's eight take away? What's first minus second? 10 to the one. Why is it alkaline? Well, look up, eight to the right is increasing alkaline. Milk of Magnesia is how many times as what as blood? Milk of Magnesia, 10.5. Human blood, 7.5. What's 10.5 take away 7.5. 10 to the third, here's blood, here's milk of Magnesia, it's to the right, so what word am I going to use? Acid or alkaline? Alkaline. See, I don't know anything about PH, but I can read a chart, alkaline. What's 100 times as acidic as normal rain? Normal rain, now acidic means head to the left, 100 means 10 to power actually, 100 times more acidic vinegar, 100 times more alkaline, pure water than normal rain, if they wanted me to go in that direction. The only one I don't like the wording of is E, E says eggs are how many times as alkaline as washing soda. See, washing soda is a 12, eggs are an eight, but they want me to go in the alkaline direction. They're saying how many times more alkaline is this than this, but this is to the left of it, it's going to be 10 to the negative 4, it's less alkaline, it's 110,000 as alkaline. Yes, hustle back, so turn the page, we're going to use 10 to the power of first minus second equals how many times more alkaline or acid depending on which way you're moving on the chart. If I give you a PH question Cassandra, I'll give you that little chart with the arrows, I'll paste that into the question. Example nine, pure water as a pH of seven, swimming pool water as a pH of 7.5, seawater as a pH of 8.5, how many times as alkaline is seawater than pure water? Seawater is 8.4, pure water is 7, is seawater to the right, is it alkaline? Why, yes it is. It's going to be 10 to the power of 8.4 minus seven, it's going to be 10 to the power of, I can go 8.4 minus seven in my head, it's 10 to the 1.4, I have no idea what that is, someone crunch that please, it's going to be bigger than 10 and smaller than 100 because 10 squared is 100 and 10 to the 1 is 10, 25? So seawater is 25 times more alkaline than pure distilled water. By the way, I think pH scale is done silly, why is seven neutral? I would have made zero neutral, acid can be negative, alkaline can be positive, but that's because a math person never made up this game. How many times is alkaline as swimming pool is seawater? So seawater is 8.4, swimming pool water 7.5, 10 to the 8.4 minus 7.5, 10 to the 0.9, 10 to the 1 is 10, so 10 to the 0.9, I'm going to get seven or eight point something, what do you get, 7.9. So seawater is 7.9 times more alkaline as swimming pool water, it's to the right, so it's alkaline, blah blah blah, we're not going to get into this in this kind of detail, you can cross out those ones. So Richter scale was 10 to the first minus second equals how many times stronger, pH was 10 to the first minus second equals how many times more acid or alkaline, look at the stupid chart and figure it out. Can you turn back please to page 213, sorry, 15, loudness of sound, loudness of sound, the decibel scale. The loudness of sound was originally measured in bells named after a Canadian, named Alexander Graham Bell, you may have heard of him, he invented the telephone. The problem is a bell is very, very big, we don't use that as our standard measurement, we use decibels as our standard measurement of sound. So a jet engine is about 140, 150 decibels, that would damage your hearing. In fact, the threshold of pain is right around 125 decibels, a whisper is around 30 decibels, you know how we compare sounds here, it's almost identical, it's going to be 10 to the power of first minus second equals how many times stronger, that would be correct Stephanie, if we measured sound in bells, but we don't measure sound in bells, we measure them in decibels, the prefix, desi, what the number does that mean, you have to divide the exponent by 10, it's 10 to the first minus second divided by 10 equals how many times, and you know how I remember, desi bells, that's my trigger phrase, like decade is 10 years, decimal is, you know, powers of 10, so a power saw is how many times as long, loud as a telephone dial, it's not 40, 10 to the fourth, not 10 to the 40th, a jet engine, 145 decibels is how many times as loud as a threshold of pain, 145 take away 125 is 20, but it's not 10 to the 20th, it's 10 to the 2, again I don't like the way C is worded, a whisper is how many times as loud as a conversation, well 30 take away 60 is negative 30, it's 10 to the negative 3, it's 11,000 as loud, turn the page, blah blah blah blah, here's your equation, 10 to the bigger, to the first one minus the second one divided by two equals, and this is their abbreviation for how many times as strong, skip example five, example six, how many times more intense is the sound of piano playing, then a whisper, well 10 to the 67 minus 22, but you got to divide that by 10, that's how many times stronger, 67 take away 22 equals divided by 10, it's going to be 10 to the power of 4.5, 31,623 times as strong, the piano is 31,623 times as loud as somebody whispering, example seven, two phones in a home ring at the same time, and each ringer has a loudness of 80 decibels, 80 dB, does that mean the total loudness is 160 decibels? Why or why not? I saw Kirsten shaking her head, she's correct, why? He turned back a page Kirsten, what was the threshold of pain? 125, when two phones ring in your house, do you drop to the ground screaming, covering your ears, hoping your fillings don't shatter because they're 160 decibels louder than a jet engine? So I know the answer is not 160 decibels, I know that I don't think you go 80 plus 80, the real question is then how do you add decibel values? This is the question in your homework I'm going to give you to see if you can figure it out before I'll talk about it next class. Speaking of homework, page 219, number one, number two, four, five, six, seven is good, skip eight, skip nine, skip ten, eleven is the question where it's going to ask you to see if you have two jets flying together at 120 decibels each. What are they combined? I'll see if you can figure it out. If not, I'll talk about it next class. Number 12, in your take home quiz. Now listen closely, listen very closely. I'm going to give you an important hint. On your test, there's going to be one or two application questions. I'm either going to ask you Richter scale, decibels, or pH? On your test, I'm either going to ask you pH, decibels, or Richter scale. If you're paying attention right now, you may have picked up a hint about what I might ask on a test. I'm telling you right now I'm going to ask you pH, Richter scale, or decibels. Take home quiz and you have homework.