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Video transcript

Use a calculator to find log base e of 67 to the nearest thousandth. So just as a reminder, e is one of these crazy numbers that shows up in nature, in finance, and all these things, and it's approximately equal to 2.71 and it just keeps going on and on and on. So you could view log base e as 67. Let's see, what does e mean? e is just a number, just like pi is just a number. So this is really the same thing as saying log base 2.71, and the actual numbers, so you'd have to write all the digits that keep on going forever and never repeat 6 of 67. So what power do I have to raise e to to get to 67? So another way of saying that is this is equal to x. You're saying e to the x is equal to 67, we need to figure out what x is. Now, traditionally you will never see someone write log base e even though e is one of the most common bases to take a logarithm of. And so the reason why you wouldn't see log base e written this way is log base e is referred to as the natural logarithm. And I think that's used because e shows up so many times in nature. So log base e of 67, another way of saying that-- or seeing that, and the more typical way of seeing that is the natural log. And I think this is ln, so I think it's maybe from French or something, log natural, of 67. So this is the same thing as log base e of 67. This is saying the exact same thing. To what power do I have to raise e to to get 67? When you see this ln, it literally means log base e. Now, they let us use a calculator, and that's good because I don't know off the top of my head what power I have to raise 2.71 so on and so forth-- what power I have to raise that to to get to 67. So we'll get our calculator out. So we get the TI85 out. And different calculators will have different ways of doing it. If you have a graphing calculator like this, you literally can literally type in the statement natural log of 67 then evaluate it. So here this is the button for ln, means natural log, log natural, maybe. ln of 67, and then you press Enter, and it'll give you the answer. If you don't have a graphing calculator, you might have to press 67 and then press natural log to give you the answer, but a graphing calculator can literally type it in the way that you would write it out, and then you would press Enter. So 4.20469 and we want to round to the nearest thousandth. So this is the thousandths place right here, this 4. The digit after that is 5 or larger, it's a 6, so we're going to round up. So this is 4.205. So this is approximately equal to 4.205. And it actually makes a lot of sense, because we know that e is greater than 2, and it is less than 3. And if you think about what 2 to the fourth power gets you to 16. And 3 to the fourth power gets you to 81. 67 is between 16 and 81 and e is between 2 and 3. So at least it feels right that's something that's like 2.71 to the little over the fourth power should get you to a number that's pretty close to 3 to the fourth power. Actually that makes sense because it's actually closer to 3. 2.71 is closer to 3 than it is to 2. So this feels right, that you take this to the fourth, little over the fourth power, you get to 67.