 This video is going to talk about applications with logs. In 1935, Charles Richter defined magnitude of an earthquake to be defined as m is equal to log times i over s, where i is the intensity of the earthquake measured in centimeters of a seismograph reading taken 10 kilometers from the center of the earthquake and s is the intensity of a standard earthquake whose amplitude is 1 micron is equal to 10 to negative 4 centimeters and a standard earthquake is the smallest measurable earthquake known. So the strongest record earthquake in 1960 was given by this intensity which is actually 3,162,277.77 centimeters and we want to know what the magnitude is. So m is equal to i which they gave us 3,162,277.77 centimeters divided by the standard earthquake which is 10 to the negative 4 and it's the log of all of this. So we come over to our calculator and say the log of 3,162,277.77 divided by 10 carat negative 4 close at parenthesis and we find out that the magnitude is equal to approximately a 10.5, 10.5 actually Richters is what they call them. And then the last one. The deadliest earthquake in 2003 was in southeast Iran and made a magnitude of 6.6. What is its intensity? Well the magnitude is 6.6 and that's equal to the log of the intensity which they're asking us for divided by that 10 to the negative 4 which we know to be the standard. So in order to solve this one we could again convert it to an exponential and the base is 10 and you hop across the pond to get to the 6.6 and then it's equal to hop back across to find out what it's equal to and that's I divided by 10 to the negative 4. And if I want to solve that I have to multiply both sides by that 10 to the negative 4 to clear the fraction. The intensity is going to be equal to 10 carat negative 4 and we're going to multiply that by 10 carat 6.6 so that the intensity is 398.107 and that would be in centimeters. So another type of logarithmic application that we use is decibels. Loudness of sound is measured in decibels and the lowest intensity sound is going to be 10 to the negative 12 watts per square meter which is what the human ear can detect. So the loudness of sound in the decimal can be found by L of I loudness of intensity is equal to 10 times the log of the intensity divided by 10 to the negative 12 when the sound of intensity is 100 watts per square meter. So a sound produced by hairdryer of intensity 10 to the negative 3 watts per square meter what is the decibel level of the dryer. So L of I is going to be equal to 10 times the log of the intensity which we know to be 10 to the negative 3 divided by 10 to the negative 12. Now this is just a plug and chug and you can stick it right into your calculator if you wanted to just like this or we do know some exponent properties so let's just remind ourselves about our exponent properties. When you have the same base and you're dividing you subtract your exponents so this is going to be 10 to the negative 3 minus a negative 12 which will actually be plus 12 or 10 times the log of 10 to the 12 minus 3 would be 9. So 10 log and then we have in parentheses 10 care at 9 or if you remember anything about properties you should know that that's going to be 10 times 9 but maybe you didn't know that so let's just say this is 90 decibels. Finally the sound of an iPod at peak volume is 115 decibels that's the L of I and we want to know what the intensity is so we're trying to solve for this I so 115 decibels is equal to 10 times the log of I which we're trying to find divided by 10 to the negative 12. So what do we do here? Well you should know that this is 10 times that log so we know that we're trying to get to the log by itself eventually so that we can convert usually logs we convert or we can take to our calculator so we have 115 divided by 10 is equal to the log of I over 10 to the negative 12 hopefully you know that this is 11.5 it goes in perfectly so 11.5 is equal to this log and now you've got some choices okay we can convert it or since it's the log we actually can graph it and we haven't graphed a log before so let's try graphing this one clear out any equations we have in there and 11.5 is going to be my Y1 and log and then I have my X divided by 10 to the negative 12 negative 12 and I should be able to graph that and let's see if we have enough to be able to graph it and it looks like we just might so second trace five intersections what we want to do when we're graphing enter enter enter and here's my graph and here's my Y equal 11.5 so obviously I was in a bigger window than a standard and my graph looks like it goes something like this and we have X is equal to 0.32 approximately okay and again if you wanted to graph if you wanted to do it the other way you'd have 10 hypercross equal sign to the 11.5 is equal to I over 10 to the negative 12 that was a 12 and there we lost our one if they earlier didn't we and then I is going to be equal to 10 to the negative 12 times 10 to the 11.5 and those exponents and you're going to end up with 10 to the it's negative 12 plus 11.5 I should be 10 to the negative 0.5 and if I take which is exactly how we use the answer intensity if you go back when we did our problem here we had 10 to the negative 3 that was our intensity but if you want to check to make sure that our graphing worked which that works to 0.32 is okay if this is really that so 10 to the negative 0.5 sure enough that's our 0.32 same answer is just a different way to get there