 This video will talk about logarithms and applications. A very common type of logarithmic application is the pH level. So f of x is equal to negative log x, where x is the ion concentration given in scientific notation. We need to remember that the solution with the pH less than 7 is called an acid, and if it's greater than 7, then we know we have a base. So this one asks us to use that formula to determine the pH of black coffee and that would be x is equal to 5.1 times 10 to the negative 5 moles per liter. And then we want to determine then is it an acid or is it a base. So we want to do f of that big long number, 5.1 times 10 to the negative 5, and that's going to be equal to the negative log of 5.1 times 10 to the negative 5. So using our calculator, we need to get out of this mode and get into our regular screen. We want negative log and we're just going to plug into that 5.1. And then this might be new for some of you, but second comma says e on the top. That's one way to write exponential. The other way to write it is I'll show you in a second. And then all you have to put here is the exponent to the negative 5th. Close the parentheses and we find out that it is 4.29. The other way to do it just to show you is negative log. And then the inverse of a log is base 10. So you could do second log and that will give you 10 to the caret. And you just put in your exponent. You just need a double set of parentheses here and press enter. And it's going to, oh, I forgot my 5.1. But that's the way you would write the 10 to the negative 5. I just forgot my 5.1. That's why I don't have the same number. So I'm going to say that this is 4.29. So the pH is equal to 4.29, which is less than 7. So we have an acid. So now we come to a magnitude of an earthquake. It's given by this function where the magnitude of intensity is equal to the log of the intensity over reference intensity. And this problem is asking us to compare the intensities of these two quakes. So we have to look at each of them separately first. So we're going to come in here and say that 9.5 is my magnitude. So 9.5 is equal to the log of i over i0. The intensities are going to be with respect to the reference. We don't know what the reference is, but we're going to put it with respect to that. So if I convert this, I have 10 to the 9.5 is equal to the i over i0. And if I want to know what the intensity of this particular earthquake is, then I would say that i is equal to 10 to the 9.5 i0. Multiplying both sides by i0 so that we can clear the fraction. And then if I come and do the other one, I would have 8.7 is equal to my log of i over i0. And then we again are going to convert it. So 10 to the 8.7 is equal to i over i0. And then when we clear the fraction, we have 10 to the 8.7 is equal to i0. Multiplying both sides by that is equal to i. So now we want to know the chilly one compared to the northern Sumatra one. We want to make that ratio. So this one is 10 to the 9.5 i0 over 10 to the 8.7 i0. The i0s are going to cancel. And when we subtract our exponents, we get 10 to the 0.8. And if we take that to your calculator, you're going to find out that that is 10 to the 0.8 is equal to 6.3. So the chilly earthquake is 6.3 times intense as the northern Sumatra. Because I don't have much space, I'm going to use a bunch of letters. So the chilly earthquake is 6.3 times more intense than the northern Sumatra. All right. So let's see what happens when we talk about the brightness of a star moving along. So here we are again. We've got this magnitude, but it's a little bit different function. But we do the same thing. 1.6 is going to be equal to 6 minus 2.5 times the log of i over i0. So you subtract the 6 and you have negative 4.4 is equal to negative 2.5 log of i over i0. And then coming back over here, I can divide negative 4.4 by negative 2.5, which is going to be positive since it's a negative divided by a negative. And that's equal to my log of the i over i0. Now we're ready to convert because we've got this as a single log all by itself. So we say that 10 to the 4.4 over 2.5 is equal to i over i0. And the intensity of this star then is going to be multiplying by i0. 10 to the 4.4 over 2.5 i0. And if we take that to our calculator, we find out that it's 57.5 times the reference intensity. So what happens when we have one that looks like this? i is equal to 80 over i0. Well, we can substitute for i here. So we have m of i is going to be equal to 6 minus 2.5 times the log of i. But now we know that i is 80 i0 over i0. And you can see that those are going to cancel. So now we have 6 minus 2.5 times the log of 80. And again, taking that to your calculator, I haven't done any with my calculator lately. So let's try again. We have 6 minus 2.5 and then log and are 80 exactly like we see it on paper. And we find out that that is going to be 1.24 or 1.2. So that would be the magnitude. It's 1.2. One last problem, memory retention. So again, we have our nice log function and we just need to know what it all represents. So p of x is the percentage of random facts retained after x number of days. So it says find the percentage of facts a person might retain after x days for the given values that many of the values are given, given our powers of 2. So we may have to use the change of base formula to get to what we want. So when it tells us that we have many values given in powers of 2, notice it is a log base 2. Then we need to change the base on that. So let's just go ahead and start there. We have log 2 of x, base 2 of x, and that would be equal to the log of the argument, which is x, over the log of 2. So anytime I see log base 2 of x, I can just replace it with log x over log 2. So for one day, we would have p of 1 is going to be equal to 95 minus 14, but I don't write the log anymore. I write the log of my x, which is 1, over the log of that base, which was 2. And if we go to the calculator for that, we say 95 minus 14. And in parentheses, we have log 1 divided by log 2, double parentheses to close what we're multiplying 14 by. And we get 95, which means that the percent of retaining those random facts for one day is going to be 95%. And so p of 4, again, same thing. Now that we've already converted that, that makes it very simple. We just say log of x, which happens to be 4 this time, over log of 2. Now don't think you can divide that 4 and 2, because remember log 4 is a number. That's not 4. And log 2 is a number. And if we plug and chug that in our calculator, we find out that that's 67%. And I want to let you do this one. So take a pause and set up the number 16, and I'll just tell you what it's equal to. And that's equal to 39%.