Logarithm Word Problems | Richter Scale and dB

hello everyone welcome back today we are going to be solving problems with exponential and logarithmic functions with a focus on word problems this time because I know last time we talked about how to solve logarithmic functions today we're going to talk about specifically with word problems what we mean by that so let's start with example one it says three earthquakes occurred in locations a b and c with magnitudes four six and seven respectively on the Richer scale how many times stronger was earthquake B than earthquake a and how many times stronger with the earthquake at Sea than the earthquake at a okay so when it comes to the Richer scale we know that this scale going up by one is actually multiplying by 10 right a multitude of 10 okay and so if we're going from A to B which is the first one from A to B well a is power is four and then B's power is six so we have to go up two right to get there right so we have five and then we have B which is at six and we're going to go up one which is really times 10 right because anything on the Richard scale going up by one is really times 10 and then times 10 again which is 10 squared or 100 so therefore B is a hundred times stronger than a and we can do the same thing with C right here we have a which is 4. and then we have B is 6. and then C is seven right so to get from four to five it's ten times to get from five to six it's ten times to get from six to seven it's ten times and so therefore we have a thousand right 10 times 10 times 10 is a thousand so therefore C is a thousand times stronger oh stronger than a awesome okay so let's move on to example two example two says if one earthquake has a magnitude of 3.5 on the Richer scale and the second earthquake has a magnitude of 6.2 compare the intensities of the two earthquakes right so they want to know basically how many times stronger is the 6.2 as compared to the 3.5 so to compare we're going to say well the one has the magnitude of 10 to the power of 6.2 and the other one has the magnitude of 10 to the power of 3.5 so that means we have 10 to the power of 6.2 minus 3.5 which is 10 to the power of 2.7 which is 501.19 approximately so therefore the second earthquake is 501.19 times stronger than the first so I think we're getting the hang of this now right that the um the Richer scale it's a scale of powers of 10 right every time I go up one on the Richer scale we're really multiplying by 10 times the intensity right every single time we go up one right right so it's it's different it's different than what we're used to because this has to do with like our logarithms our exponents okay so what about example three it says according to the table given out in class Amplified rock music has a loudness of 120 decibels an ordinary conversation has a loudness of 50 decibels and a whispering has a conversation of 20 decibels how many times louder is an ordinary conversation than whispering okay well when it comes to the loudness scale going up 10 on the loudness scale is multiplying by 10 in terms of intensity right so if we have a loudness of whispering so whispering is 20 right is 20 decibels then we have 30 40 and then 50 is the ordinary conversation each time we're going to go up by 10. so let's write that in so here from here to here it's times 10 from here to here it's times 10 and from here to here it's times 10. so that's a thousand right 10 times 10 times 10 is a thousand so therefore uh ordinary conversation is a thousand times louder than a whisper okay well I mean at least according to this I don't know how accurate these are I'm not a science person but assuming that's accurate that's that's kind of crazy a thousand times more than a whisper okay let's do B which says how many times louder is a rock concert than ordinary conversation so an ordinary conversation is 50 then we have 60 70 80 90 100 110 and then finally 120. for the rock concert so each of these are times tens we have times 10. times 10 times 10 times 10 times 10 times 10 and times ten so one two three four five six seven so seven so 10 to the power of 7 times louder so ordinary conversation or I guess I could say rock concert is 10 to the power of 7 times louder than the ordinary conversation good all right example four says the sound of a person screaming is 104 decibels the noise level of the same person shouting is 87 decimals how many times louder is the screen over the shouting of this person sounds like they can scream louder than they can shout which is interesting okay so to compare in this case we're going to do 10 to the power of 104 over 10 right because in this case to go up 10 we multiply by 10 as opposed to going up one we multiply by 10. so that's why we have to divide by 10 here right as opposed to a richer scale where we didn't do this because going up by one on the Richard scale is like multiplying by 10. and that's going to be over the screaming which is 8 87 over 10. so that is 10 to the power of 10.4 over 10 to the power of 8.7 which is 10 to the power of 1.7 which is 50.12 so therefore the shouting or I should say screaming because screaming is the louder one screaming for this person at least is 50.12 times louder than shouting which that that's interesting I would have thought it wouldn't have been that much but that's a lot good okay very good okay let's move on to this chemistry example in chemistry the pH the measure of the acidity or alkalinity of a substance is based on the logarithmic scale using the formula pH equals negative log h plus where H plus is the concentration of hydrogen ions in moles per liter a says find the pH level of a substance if the hydrogen concentration is 0.00001 moles per liter okay so in this case we know we just have to plug this back in we know that pH is going to be equal to negative 0.000 there's one two three four five zeros one um moles per liter oh sorry I forgot to put log it should be negative log and then h plus 0.00001 moles per liter and then what would that be well 10 to the power of what is going to give us 0.00001 well we'd have to move this decimal point over one two three four five six seven places right in order to get back to 10. right so that's a lot of places 10 places One Two Three or Sorry Seven I think why do I say ten I meant seven I think yeah because we'd have to go one two three four five six seven to get it back to just regular ten so ten to the power of negative seven therefore would be zero point zero zero zero zero one so therefore the answer is negative seven and therefore the pH would be seven because minus negative seven is just going to give us regular seven then it asks find the hydrogen concentrate level of a substance that has pH of 9.3 okay so in that case what we'll do is we'll put 9.3 for pH and we'll put over here negative log of h plus and that's what we're trying to figure out okay so first things first let's put the negative on the other side so negative 9.3 equals log h plus and now let's change this into exponential form 10 to the power of negative 9.3 is going to equal to the hydrogen Plus and well that's it that's the answer we could do that out and say that h plus is equal to 0.0005 if we wanted all right because one two three four five six seven eight nine we could do that if if we really wanted to do so but this answer is perfectly valid too so I'm just going to leave it like that I think it's much nicer to write it that way very good all right and then example six says the amount of water vapor uh in the air is a function of temperature given by the equation s equals 5.06 times 1.07 to the power of 0.95 T where s is the saturation of the air in milliliters per meters cubed and T is the temperature in degrees celsius find the temperature of the air if the saturation is 50 milliliters per meter cubed okay so to find the temperature we want to find T if s is 50. okay so let's plug that in that 50 equals 5.06 1.07 0.95 T like that okay and then let's divide by 5.06 9.88 is equal to 1.07 to the power of 0.95 t all right and then we're going to log both sides so we're going to say log 9.88 is equal to log 1.07 0.95 T because now what we can do is we can put the 0.95 T at the front right as the coefficient because that's one of our logarithm laws we talked about a couple days ago so this is the same as 0.95 T Times log 1.07 and that's equal to log 9.88 so therefore 0.95 t is equal to log 9.88 over log 1.07 good all right and what does that mean for us um let's see so that means T would be what T would be about 35.5 degrees Celsius and there we have it awesome okay next one is find the temperature of the air if the saturation so if s is five milliliters per meter cubed so instead of fifth so instead of 50 this time we're just going to do 5. okay all right so let's do that so that means 5 is equal to 5.06 times 1.07 to the power of 0.95 T so if we divide both sides by 5.06 we're going to get 0.99 equals 1.07 to the power of 0.95 t okay then we're going to log both sides so we're going to get log 0.99 equals log 1.07 to the power of 0.95 t and then we'll put that 0.95 t out front so 0.95 t log 1.07 is equal to log 0.99 and then so therefore 0.95 T is equal to log 0.99 over log 1.07 okay so therefore T would be oh around zero I guess negative 0.16 degrees Celsius is a bit more accurate though so we'll go with that all right well there we go congratulations on completing this uh unit oh and I should mention here here is actually a copy of the decibel scale and just some things we could kind of compare it to and then also the Richard scale here's a copy of the Richard scale as well but yeah that is it for stay good works day everyone and I will see everybody in the next unit bye everyone