Logarithms | Logarithms | Algebra II | Khan Academy
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Logarithms | Logarithms | Algebra II | Khan Academy


Let’s learn a little bit about the wonderful world of logarithms. So we already know how to take exponents. If I were to say 2 to the fourth power, what does that mean? Well that means 2 times 2 times 2 times 2. 2 multiplied or repeatedly multiplied 4 times, and so this is going to be 2 times 2 is 4 times 2 is 8, times 2 is 16. But what if we think about things in another way. We know that we get to 16 when we raise 2 to some power but we want to know what that power is. So for example, let’s say that I start with 2, and I say I’m raising it to some power, what does that power have to be to get 16? Well we just figured that out. ‘X’ would have to be 4. And this is what logarithms are fundamentally about, figuring out what power you have to raise to, to get another number. Now the way that we would denote this with logarithm notation is we would say, log, base– actually let me make it a little bit more colourful. Log, base 2– I’ll do this 2 in blue… Log, base 2, of 16 is equal to what, or is equal in this case since we have the ‘x’ there, is equal to ‘x’? This and this are completely equivalent statements. This is saying “hey well if I take 2 to some ‘x’ power I get 16′.” This is saying, “what power do I need to raise 2 to, to get 16 and I’m going to set that to be equal to ‘x’.” And you’ll say, “well you have to raise it to the fourth power and once again ‘x’ is equal to 4. So with that out of the way let’s try more examples of evaluating logarithmic expressions. Let’s say you had… log, base 3, of 81. What would this evaluate to? Well this is a reminder, this evaluates to the power we have to raise 3 to, to get to 81. So if you want to, you can set this to be equal to an ‘x’, and you can restate this equation as, 3 to the ‘x’ power, is equal to 81. Why is a logarithm useful? And you’ll see that it has very interesting properties later on. But you didn’t necessarily have to use algebra. To do it this way, to say that ‘x’ is the power you raise 3 to to get to 81, you had to use algebra here, while with just a straight up logarithmic expression, you didn’t really have to use any algebra, we didn’t have to say that it was equal to ‘x’, we could just say that this evaluates to the power I need to raise 3 to to get to 81. The power I need to raise 3 to to get to 81. Well what power do you have to raise 3 to to get to 81? Well let’s experiment a little bit, so 3 to the first power is just 3, 3 to the second power is 9, 3 to the third power is 27, 3 to the fourth power, 27 times 3 is equal to 81. 3 to the fourth power is equal to 81. ‘X’ is equal to four. So we could say… Log, base 3, of 81, is equal to– I’ll do this in a different colour. Is equal to 4. Let’s do several more of these examples and I really encourage you to give a shot on your own and hopefully you’ll get the hang of it. So let’s try a larger number, let’s say we want to take log, base 6, of 216. What will this evaluate to? Well we’re asking ourselves, “what power do we have to raise 6 to, to get 216?” 6 to the first power is 6, 6 to the second power is 36, 36 times 6 is 216. This is equal to 216. So this is 6 to the third power is equal to 216. So if someone says “what power do I have to raise 6 to– this base here, to get to 216?” Well that’s just going to be equal to 3. 6 to the third power is equal to 216. Let’s try another one. Let’s say I had, I dunno, log, base 2, of 64. So what does this evaluate to? Well once again we’re asking ourselves, “well this will evaluate to the exponent that I have to raise this base 2, and you do this as a little subscript right here. The exponent that I have to raise 2 to, to get to 64.” So 2 to the first power is 2, 2 to the second power is 4, 8, 16, 32, 64. So this right over here is 2 to the sixth power, is equal to 64. So when you evaluate this expression you say “what power do I have to raise 2 to, to get to 64?” Well I have to raise to the sixth power. Let’s do a slightly more straightforward one, or maybe this will be less straightforward depending on how you view it. What is log, base 100, of 1? I’ll let you think about that for a second. 100 is a subscript so it’s, log, base 100, of 1. That’s one way to think about it, I’ll put parentheses around the 1. What does this evaluate to? Well this is asking ourselves, or we would evaluate this as, “what power do I have to raise 100 to, to get to 1?” So let me write this down as an equation. If I set this to be equal to ‘x’, this is literally saying 100, to what power, is equal to 1? Well anything that a 0 power is equal to 1. So in this case ‘x’ is equal to 0. So log, base 100, of 1, is going to be equal to 0. Log base anything of 1, is going to be equal to 0 because anything to the 0 power and we’re not talking about 0 here. Anything that is to the power of 0 that is not 0, is going to be equal to 1.

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100 thoughts on “Logarithms | Logarithms | Algebra II | Khan Academy

  1. Holy Crap thank you so much, you're a really good teacher, and color coding it was very helpful, I couldn't undertand it cause my teacher made it sound so complicated. Thanks again, for sharing!

  2. I'm sad that this was so simple. I really wish I had found this specific video when I was taking algebra in high school. I actually still couldn't remember logarithms (nor did I understand it at all). You made this very simple. My teacher was horrible the way she explained it and the way the class was taught was that we were put in groups. In high school, groups don't get ANYWHERE in learning. I hope I made sense lol

  3. Thats high school in general, i learned jack there because of the way they taught it. they just recycle the same shitty was to explain it on the way they taught it and if you don't get it then thats it you miss out. its nice now that we have plenty of options to learn.

  4. Is there any way to solve logs without a calculator. I'm thinking that there must be. And I'm not talking about something like log base 2 of 8 or log base 2 of 16. Those are simple cuz the answers to those are whole numbers: 3 and 4. I'm talking about something like: log of 2. The answer to that is about 0.301… I solved this through a calculator. How can I do it from scratch?

  5. This man is a saint.  What a wonderful time we live in where a guy like Mr. Khan is kind enough to share his intellect and teaching ability on the miracle that is the internet so people like me can learn in 2 minutes what an entire expensive 3 hour college algebra lecture failed to do.  Thank you, sir.  It's people like you that make the world go 'round.

  6. maaan, I always used to think about logarithms as of somethings mysterious and completely incomprehensible, but 2 minutes of this this video completely changed it

  7. I'm a homeschooled student who's always had problems with math,but the kahn Academy site and videos have really helped me improve and actually like math(something I thought I'd never say!). To the creator of Kahn Academy,thank you.

  8. wtf i dont want to do no guess and test method, teach me how to solve completely random ones like log base 3 (1230193013)

  9. Wow I should pay you to be my teacher. I have a really shitty math teacher as a senior and I have suffered pretty much all year because of this crappy teacher who can't teach. She just skips most of the steps and whenever you go up and ask something specific that requires prior knowledge(hello, people forget..), she just says you should have learned this from last year and then she refuses to help. My pre-calc teacher needs to be fired! She is horrible!

    Thanks Mr. Khan/Sal you are an amazing teacher. You should make money off of this because you are helping so many people around the world in a few minutes whereas shitty math teacher can't over the course of an year

  10. can someone please explain how 16 is the power? Other sites are saying the 4 would be the power–or the exponent, but nobody is assigning the power to the number in the 16's position. It makes no sense to me…but i admit I am struggling with this.

    This is from the intro to logarithms text page after the video.
    24=16⟺log2(16)=4
    Both equations describe the same relationship between the numbers 2, 4, and 16 where 2, is the base, 4 is the exponent, and 16 16 is the power.

  11. Is there an easier way to evaluate logarithmic expressions WITHOUT creating some sort of factorization tree or running through a "guess and check" / "trial and error" method?

    For instance, there are defined processes with guidelines to add, subtract, multiply, and divide large numbers. How would I quickly evaluate a logarithmic expression that I do not have "memorized"?

    This video is one of the more popular YouTube video on logarithms, so please upvote this comment to let all of the wandering math majors and wizards see this comment.

  12. https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/introduction-to-logarithms/v/logarithms

  13. Holy fuck dude? i understood in 7 mins , took me 3 years in high school struggling to understand wtf is log , i blame stupid teachers who work inefficiency for money

  14. not gonna lie, you got me on the last one.. great lesson.. I will say when you said, ok let's try another one, I was like NOOOOOOOOO:) I know you have to practice and take notes to get any subject.. math is no different.. thanks again!!

  15. I'm in 8th grade and man there still teaching how to solve inequalities and equations

    Like damn they think we're stupid

    This is easy af and I'm sure teachers in my junior year (future) would take 5 months reviewing algebra 1 and the rest of the year doping around

  16. Thank you so much! I have my exam soon and our teacher hated every one of us and explained absolutely terribly. I didn't understand this chapter throughout the entire year and you've finally helped me understand it IN 7 MINUTES. Our teacher couldn't do it in a year!

  17. I got stuck on logarithm in high school, now my youngest son in 4th grade knows about it. I always had to just turn into the back of the book to find the logarithm, not understanding anything I was doing. Wow…simple and precise. Thanks Mr. Kahn.

  18. why the hell this person tells how it works in a simple basic words while teachers are saying some mysthical words we forget

  19. I’m failing algebra 2 😭 hopefully I can learn everything online bc I’m currently not able to learn anything in class.

  20. I was interested in learning this after my teacher in alg 1 class used his calculator and it showed a bunch of logs… actually seems pretty simple

  21. I still don't understand what the log button on my calculator does, when I type log (2)16 I get 4.8. Also what is happening when I just push log (5) = or any other number??? What happens when I log transform a data set?

  22. Thanks for the video! Needed a little refresher and this brought my previous knowledge back. Khan Academy is amazing, thanks for making so much of math simple and understandable.

  23. At 1:40 they are not totally equivalent statements at all. They might contain the same components but they are not equivalent.

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