# Introduction to logarithm properties (part 2) | Logarithms | Algebra II | Khan Academy

PROFESSOR: 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, that 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 3

times logarithm base 2 of 8. So this property tells us that

this is going to be the same thing as logarithm base 2

of 8 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. 3 times log base–

what’s log base 2 of 8? 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? 2 to the what power is 8? 2 to the third

power is 8, right? So that’s 3. We have this 3 here,

so 3 times 3. So this thing right

here should equal 9. If this equals 9, 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 2

to the ninth power? Well it’s going to be

some large number. Actually, I know what

it is– it’s 256. Because in the last video we

figured out that 2 to the eighth was equal to 256. So 2 to the ninth

should be 512. So 2 to the ninth

should be 512. So if 8 to the third is also

512 then we are correct, right? Because log base 2 of 512

is going to be equal to 9. What’s 8 to the third? It’s 64– right. 8 squared is 64, so 8

cubed– let’s see. 4 times 2 is 3. 6 times 8– looks

like it’s 512. Correct. And there’s other ways

you could have done it. Because you could have said

8 to the third is the same thing as 2 to the ninth. How do we know that? Well, 8 to the third is

equal to 2 to the third to the third, right? I just rewrote 8. And we know from our exponent

rules that 2 to the third to the third is the same

thing as 2 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 17 of 357. And you will scramble and look

for the log base 17 button on your calculator,

and not find it. Because there is no log base

17 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 10. 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 nothing– 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 10. We could say that’s the same

thing as log base 10 of 357 divided by log base 10 of 17. So you literally could just

type in 357 in your calculator and press the log button and

you’re going to get bada bada bam. Then you can clear it, or if

you know how to use the parentheses on your calculator,

you could do that. But then you type 17 on your

calculator, press the log button, go to bada bada bam. 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 let’s just

rewrite a bunch of things in simpler forms. So if I wanted to rewrite the

log base 2 of the square root of– let me think of something. Of 32 divided by the cube– no,

I’ll just take the square root. Divided by the

square root of 8. 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 2 of 32 over the square root of 8 to

the 1/2 power, right? And we know from our logarithm

properties, the third one we learned, that that is the same

thing as 1/2 times the logarithm of 32 divided by the

square root of 8, 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 32 divided by

logarithm of square root of 8. Well, we can use our

other logarithm– let’s keep the 1/2 out. That’s going to equal,

parentheses, logarithm– oh, I forgot my base. Logarithm base 2 of

32 minus, right? Because this is in a quotient. Minus the logarithm base 2

of the square root of 8. Right? Let’s see. Well here once again we have a

square root here, so we could say this is equal to 1/2

times log base 2 of 32. Minus this 8 to the 1/2,

which is the same thing is 1/2 log base 2 of 8. We learned that property in the

beginning of this presentation. And then if we want, we can

distribute this original 1/2. This equals 1/2 log base 2 of

32 minus 1/4– because we have to distribute that 1/2–

minus 1/4 log base 2 of 8. This is 5/2 minus, this is 3. 3 times 1/4 minus 3/4. Or 10/4 minus 3/4

is equal to 7/4. I probably made some arithmetic

errors, but you get the point. See you soon!

Was khan on his man period while making this video?

@goldensilverstar Can't tell, but I have to use it to calculate frequency table in my classes of probability at the university. I'm studying Computer Science, and I forgot some principles…cough! … to be blunt, I forgot everything about logarithms. There's some rumors I'll use it also in circuit classes. This I'll see, soon.

Wow. Ok // I want to study computer science too. Software Engineering.

wait how is log base2 (sqrt32/sqrt8) = log base2 (32/sqrt8)^(1/2)? shouldn't it only be 32^(1/2)? like (32^(1/2)/sqrt8) isnt that right? wtf?

oh its because he actually had log base2 sqrt(32/sqrt8) lol nevermind

I often find myself sitting in class watching what's being written on the board and think " when will i ever need this?!"

after watching 2 of your videos i can actually see myself using logarithms in a real life situation and not just on a test to get a grade for school 😀

thanks man 😉

@4:15…. Lol I do that all the time, I will say what I'm going to write, and then just sit there stroking the pencil in mid-air… I hate when I do that….. Lol He does it twice at 8:27 too!

almost missed this video cuz I didn't see your black background in the thumbnail lol

I was not a fan of government-funded cloning programs . . . until Khan showed up. *cue Imperial Theme*

this is not fair! y are u not my maths and science teacher??

lol you say"i don't know"

part duex

I really liked the "don't feel bad if it's not obvious"

great videos… thanks a lot

My calculator lets me set the base but I still use the 2nd property shown (ti nspire cx)

I didn't get the last part clearly. The way the problem is written at the beginning of the last example is easy to enter into EXCEL. "=LOG(SQRT(32/SQRT(8)),2)". Other then getting an A for an exam, why would one need to know all the rules you used to solve this problem?

Sal is like literally the smartest guy on the entire fricken' planet!

My teacher said whoever passes the pre-test gets 3 weeks out off class

This guy has a really good voice for explaining things. //

Haha! I love the confidence inference:D this made learning fun. I skipped a week of school and thank heavens for khan academy(:

3log2(8) is 9, why did 512 come out?

you are a genius. thanks for the help

Can you just be my tutor for the rest if my school life? Why don't teachers explain it as simply as this???

I LOVE YOU!!

"fascinating"

please consider becoming a math teacher you could be making bank x-)

I wish i had realized that years ago. But its not too late to apply and try to understand it i guess. Being a info systems major, I think im required to understand it lol

First u would have to change to your yellow color to act with confidence XD 6:00

where in South Africa are you from?

I need to find my yellow color again now.

Thank you! God bless you! :))

I cant take it anymore. You are actually quite effective in communicating the material. however, YOU DIGRESS (A LOT) FROM THE POINT/SUBJECT so much that it is kind of confusing. Also to add to the confusion is the "machine" you are using. However, thanks for the free material it did help a lot!

garbage.

but it sure as hell helps.

Thanks a bunch for these tutorials, they were the best I could find. And since I missed my theory lessons of logarithms, this was very useful! I now begin to understand it better. Cheers!

the 240p strikes again

Thank u so much….:).

To do log with a different base on a graphing calculator hit the math button then scroll down till you see logBASE( to give the log a different base

I dont the reason of using ten for the base

i can't wait to see how you teach me the quotient rule… holy shit.

i don't understand the part where you change the 1/2 to 1/4 why you did that? can explain please and thanks

how do you get 7/4?

Actually you can do (math) (alpha) and that gives you log_ ( ) but this only works on ti 84s

retro khan academy, dayuuummmmmmmmmm!

I just do NOT understand why he kept the 1/2 out of parentheses. This would have never intuitively occurred to me if I were doing this. It seems to be completely arbitrary.

how does 1/2 cancel out a square root?

that went from easy to hard in 7 minutes and 10 seconds :/

Americans complaining about this being hard.

Sal seems much more witty in the early days! Wish he could still be like that…

I'm very confused

I'll explain how he got to convert log base 2 (32) and log base 2 (8). long base 2 (32) is 2 to the X equals 32. 32 can be divided into 8 times 4. Each of these can be factored into 2 to the 3rd and 2 to the second respectively. That will make log base 2 (32) equal to 5. then you multiply 5 with the coefficient 1/2 to get 5/2. Similarly, log base 2 (8) is equal to 3, then multiply it with 1/4 to get 3/4. You add 1/2 to 3/4 and you get 7/4.

how did he get (5/2)-(3/4)?

Sounds like the guy from vlad tv

"the point of math is to understand math"

thank you Sal! 🙂

This was good but the last question couldve been solved easily by taking 32 as 2^5 and 8 as 2^3 and then multiplying them with 1/2 coz of the sq root….or maybe thats just coz im an Indian 😛

There is a key on the calculator to get any base you want on Casio 991es plus.

after getting 9 in the first problem, I got lost with the process of getting the final answer, 512, why did we do 2 ^9?

How do you know that c is 10?

omg. tht last question was messy

logarithms r so hard for me

&(&most scientific calculators have log buttons for arbitrary bases, though.

The last rule is kind of confusing but everything else is easy about it.

"you have to switch to your yellow color to act with confidence" HAHA

I wish I could write brughbugghagahh as the answer

love you sal

i got it

"the purpose of math isn't just to pass your next test, but to understand math" No, passing my test is the only reason I'm learning this. I missed the entire unit of Logarithms because I was sick, and here I am 2 months after that unit, trying to learn what I missed so I can take the bloody test that the rest of my class took 2 months ago, so I don't fail.

i dont feel like he did a good job of explaining in this video

Why did he make the exponent into a fraction in the last logarithmic equation?

i have three bases on my calculator. The two you mentioned in the video and I also have an option to pick which base I like.

Isn't the answer suppose to be 1 not 7/4

At 8:15 wouldn't the root 8 just become 8 when you put it to the power of a half. Same as the root 2 became 32? Log base 2 of (root 32 /root 8) = 1. Log base 2 of (32/root 8)1/2 power = 1.75. Correct me if I'm wrong. Can't see how you can cancel the root on the top half of the fraction and put 32 to the power of a half but leave the bottom half with the root and still put it to the power of a half

But you get the point…

I do?

In the second property, do we need to take base as 10 in every case???

Where did 9 come fromm

"type in 357 in your calculator press the log button and you get nanana"

for some reason this made me laugh

Why did he randomly jump to 2^9 with no explanation?

what mic do u use and what screen recorder

Thank you so much for making these kind of videos their really helpful!

You lost me when you got 5/2 where did it come from?

Where's the video that explains more formally the multiplication of logarithms? He says it right at about 3:55. where's that video?

I felt like he gave up explaining at the end…

My head literally hurts rn

slow af

Thanks for your explaination sir ..

You explained in a very beautiful & understanding manner

He was hitting that 10 minute marker so he just gave up on explaining at the end 😂😂

If you don’t believe why this is true you can go watch the proof. Nope you could say pigs fly and I wouldn’t question it

Of math seems hard Ur doin it wrong

A decade later and I still rely on you. Thank you for everything Sal.

Don't understand just feel…

oof that high quality 1080 p video is great

Great video and the example at the end was superb to drive home the learning points!!

Calculators are not allowed here in India, not even in the universities and colleges.

Finally i will have something to understand in my tuition today.

Thank you so much!❤

God bless Khan academy for making studies very simple

Years later in the work force, i ended up needing to relearn this. I should of listened.

Almost 5 million subscribers. :O

what app is this in which u r writting

My calculator has a log_(anything) button, what now Sal?

ummmmm Yall please pray for me and logs

Thank you so much for everything

Where did he get 34? Isn't that suppose to be 32?