Topic page
Logarithm properties
- Introduction to logarithm properties
- Introduction to logarithm properties (part 2)
- Logarithm of a Power
- Sum of Logarithms with Same Base
- Using Multiple Logarithm Properties to Simplify
- Operations with logarithms
- Change of Base Formula
- Proof: log a + log b = log ab
- Proof: log_a (B) = (log_x (B))/(log_x (A))
- Proof: A(log B) = log (B^A), log A - log B = log (A/B)
- Logarithmic Equations
- Solving Logarithmic Equations
- Solving Logarithmic Equations
- Logarithmic Scale
- Richter Scale
Introduction to logarithm properties (part 2) Second part of the introduction to logarithm properties.
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- Well, welcome back! I'm going to show you the last two logarithm properties now.
- So this one--
- and I always found this one to be in some ways the most obvious one.
- But don't feel bad if it's not obvious.
- Maybe it'll take a little bit of introspection.
- And I encourage you to really experiment with all these logarithm properties,
- because that's the only way that you'll really learn them.
- And the point of math isn't just to pass the next exam,
- or to get an A on the next exam.
- The point of math is to understand math
- so you can actually apply it in life later on
- and not have to relearn everything every time.
- So the next logarithm property is,
- if I have A times the logarithm base B of C, if I have A times this whole thing,
- then that equals logarithm base B of C to the A power.
- Fascinating.
- So let's see if this works out.
- So let's say if I have three times logarithm base two of eight.
- So this property tells us that this
- is going to be the same thing as logarithm base two of eight to the third power.
- And that's the same thing.
- Well, that's the same thing as-- we could figure it out.
- So let's see what this is.
- three times log base-- what's log base two of eight?
- The reason why I kind of hesitated a second ago is
- because every time I want to figure something out,
- I implicitly want to use log and exponential rules to do it.
- So I'm trying to avoid that.
- Anyway, going back.
- What is this?
- two to the what power is eight?
- two to the third power is eight, right?
- So that's three.
- We have this three here, so three times three.
- So this thing right here should equal nine.
- If this equals nine,
- then we know that this property works at least for this example.
- You don't know if it works for all examples,
- and for that maybe you'd want to look at the proof we have in the other videos.
- But that's kind of a more advanced topic.
- But the important thing first is just to understand how to use it.
- Let's see, what is two to the ninth power?
- Well it's going to be some large number.
- Actually, I know what it is-- it's two hundred and fifty-six.
- Because in the last video we figured out that two to the
- eighth was equal to two hundred and fifty-six.
- So two to the ninth should be five hundred and twelve.
- So two to the ninth should be five hundred and twelve.
- So if eight to the third is also five hundred and twelve then we are correct, right?
- Because log base two of five hundred and twelve is going to be equal to nine.
- What's eight to the third?
- It's sixty-four-- right.
- eight squared is sixty-four, so eight cubed-- let's see.
- four times eight. so, it's two and three.
- six times eight-- looks like it's five hundred and twelve.
- Correct.
- And there's other ways you could have done it.
- Because you could have said eight to the third
- is the same thing as two to the ninth.
- How do we know that?
- Well, eight to the third
- is equal to two to the third to the third, right?
- I just rewrote eight.
- And we know from our exponent rules that two to the third to the third
- is the same thing as two to the ninth.
- And actually it's this exponent property, where you can multiply--
- when you take something to exponent and then take that to an exponent,
- and you can essentially just multiply the exponents--
- that's the exponent property that actually leads to this logarithm property.
- But I'm not going to dwell on that too much in this presentation.
- There's a whole video on kind of proving it a little bit more formally.
- The next logarithm property I'm going to show you--
- and then I'll review everything and maybe do some examples.
- This is probably the single most useful logarithm property if you are a calculator addict.
- And I'll show you why.
- So let's say I have log base B of A
- is equal to log base C of A divided by log base C of B.
- Now why is this a useful property if you are calculator addict?
- Well, let's say you go class, and there's a quiz.
- The teacher says, you can use your calculator,
- and using your calculator I want you to figure out the log base seventeen of three hundred and fifty-seven.
- And you will scramble and look for the log base seventeen button on your calculator,
- and not find it.
- Because there is no log base seventeen button on your calculator.
- You'll probably either have a log button
- or you'll have an ln button.
- And just so you know, the log button on your calculator
- is probably base ten.
- And your ln button on your calculator
- is going to be base e.
- For those you who aren't familiar with e, don't worry about it
- but it's 2.71 something something.
- It's a number.
- It's an amazing number, but we'll talk more about that in a future presentation.
- But so there's only two bases you have on your calculator.
- So if you want to figure out another base logarithm,
- you use this property.
- So if you're given this on an exam,
- you can very confidently say, oh, well that is just the same thing as--
- you'd have to switch to your yellow color in order to act with confidence--
- log base-- we could do either e or ten.
- We could say that's the same thing as log base ten of three hundred and fifty-seven
- divided by log base ten of seventeen.
- So you literally could just type in three hundred and fifty-seven in your calculator
- and press the log button
- and you're going to get blah blah blah.
- Then, you know, you can clear it,
- or if you know how to use the parentheses on your calculator, you could do that.
- But then you type seventeen on your calculator,
- press the log button, you get blah blah blah.
- And then you just divide them, and you get your answer.
- So this is a super useful property for calculator addicts.
- And once again, I'm not going to go into a lot of depth.
- This one, to me it's the most useful,
- but it doesn't completely--
- it does fall out of, obviously, the exponent properties.
- But it's hard for me to describe the intuition simply,
- so you probably want to watch the proof on it,
- if you don't believe why this happens.
- But anyway, with all of those aside,
- and this is probably the one you're going to be using the most in everyday life.
- I still use this in my job.
- Just so you know logarithms are useful.
- Let's do some examples.
- Let's just rewrite a bunch of things in simpler forms.
- So if I wanted to write the log base two of the square root of--
- let me think of something.
- Of thirty-two divided by the cube-- no, I'll just take the square root.
- Divided by the square root of eight.
- How can I rewrite this so it's reasonably not messy?
- Well let's think about this.
- This is the same thing, this is equal to--
- I don't know if I'll move vertically or horizontally.
- I'll move vertically.
- This is the same thing as the log base two of thirty-two
- over the square root of eight to the one half power, right?
- And we know from our logarithm properties, the third one we learned,
- that that is the same thing as one half
- times the logarithm of thirty-two divided by the square root of eight, right?
- I just took the exponent
- and made that the coefficient on the entire thing.
- And we learned that in the beginning of this video.
- And now we have a little quotient here, right?
- Logarithm of thirty-two divided by logarithm of square root of eight.
- Well, we can use our other logarithm--
- let's keep the one half out.
- That's going to equal, parentheses, logarithm--
- oh, I forgot my base.
- Logarithm base two of thirty-two minus, right?
- Because this is in a quotient.
- Minus the logarithm base two of the square root of eight.
- Right?
- Let's see.
- Well here once again we have a square root here,
- so we could say this is equal to one half times log base two of thirty-two.
- Minus this eight to the one half,
- which is the same thing as one half log base two of eight.
- We learned that property in the beginning of this presentation.
- And then if we want, we can distribute this original one half.
- This equals one half log base two of thirty-two minus one fourth--
- because we have to distribute that one half--
- minus one fourth log base two of eight.
- This is five halves minus, this is three.
- three times one fourth minus three fourths.
- Or ten fourths minus three fourths is equal to seven fourths.
- I probably made some arithmetic errors, but you get the point.
- See you soon!
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Answer more questions
Visit Community Questions to answer new questions in Logarithm properties.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.