Welcome to this presentation

on logarithm properties. Now this is going to be a

very hands-on presentation. If you don’t believe that one

of these properties are true and you want them proved, I’ve

made three or four videos that actually prove

these properties. But what I’m going to do

is I’m going to show you the properties. And then show you how

they can be used. It’s going to be

little more hands-on. So let’s just do a little

bit of a review of just what a logarithm is. So if I say that a– Oh

that’s not the right. Let’s see. I want to change–

There you go. Let’s say I say that

a– Let me start over. a to the b is equal to c. So if we– a to the b

power is equal to c. So another way to write this

exact same relationship instead of writing the exponent, is

to write it as a logarithm. So we can say that the

logarithm base a of c is equal to b. So these are essentially

saying the same thing. They just have different

kind of results. In one, you know a and b and

you’re kind of getting c. That’s what exponentiation

does for you. And the second one, you know

a and you know that when you raise it to some

power you get c. And then you figure

out what b is. So they’re the exact same

relationship, just stated in a different way. Now I will introduce you

to some interesting logarithm properties. And they actually just fall

out of this relationship and the regular exponent rules. So the first is that the

logarithm– Let me do a more cheerful color. The logarithm, let’s say, of

any base– So let’s just call the base– Let’s

say b for base. Logarithm base b of a plus

logarithm base b of c– and this only works if we

have the same bases. So that’s important

to remember. That equals the logarithm

of base b of a times c. Now what does this mean

and how can we use it? Or let’s just even try it

out with some, well I don’t know, examples. So this is saying that– I’ll

switch to another color. Let’s make mauve my–

Mauve– I don’t know. I never know how to

say that properly. Let’s make that my

example color. So let’s say logarithm of base

2 of– I don’t know –of 8 plus logarithm base 2 of– I

don’t know let’s say –32. So, in theory, this should

equal, if we believe this property, this should equal

logarithm base 2 of what? Well we say 8 times 32. So 8 times 32 is

240 plus 16, 256. Well let’s see if that’s true. Just trying out this number and

this is really isn’t a proof. But it’ll give you a little bit

of an intuition, I think, for what’s going on around you. So log– So this is– We

just used our property. This little property that

I presented to you. And let’s just see

if it works out. So log base 2 of 8. 2 to what power is equal to 8? Well 2 to the third power

is equal to 8, right? So this term right here,

that equals 3, right? Log base 2 of 8 is equal to 3. 2 to what power is equal to 32? Let’s see. 2 to the fourth power is 16. 2 to the fifth power is 32. So this right here is 2 to

the– This is 5, right? And 2 to the what power

is equal to 256? Well if you’re a computer

science major, you’ll know that immediately. That a byte can have

256 values in it. So it’s 2 to the eighth power. But if you don’t know that, you

could multiply it out yourself. But this is 8. And I’m not doing it just

because I knew that 3 plus 5 is equal to 8. I’m doing this independently. So this is equal to 8. But it does turn out that

3 plus 5 is equal to 8. This may seem like magic to

you or it may seem obvious. And for those of you who it

might seem a little obvious, you’re probably thinking, well

2 to the third times 2 to the fifth is equal to 2 to

the 3 plus 5, right? This is just an exponent rule. What do they call this? The additive exponent

prop– I don’t know. I don’t know the

names of things. And that equals 2 to

8, 2 to the eighth. And that’s exactly what

we did here, right? On this side, we had 2

the third times 2 to the fifth, essentially. And on this side, you have

them added to each other. And what makes the logarithms

interesting is and why– It’s a little confusing at first. And you can watch the proofs

if you really want kind of a rigorous– my proofs

aren’t rigorous. But if you want kind of

a better explanation of how this works. But this should hopefully give

you an tuition for why this property holds, right? Because when you multiply

two numbers of the same base, right? Two exponential expressions

of the same base, you can add their exponents. Similarly, when you have the

log of two numbers multiplied by each other, that’s

equivalent to the log of each of the numbers added

to each other. This is the same property. If you don’t believe me,

watch the proof videos. So let’s do a– Let me show

you another log property. It’s pretty much the same one. I almost view them the same. So this is log base b of

a minus log base b of c is equal to log base b

of– well I ran out. I’m running out of space

–a divided by c. That says a divided by c. And we can, once again, try

it out with some numbers. I use 2 a lot just because

2 is an easy number to figure out the powers. But let’s use a

different number. Let’s say log base 3 of– I

don’t know –log base 3 of– well you know, let’s make it

interesting –log base 3 of 1/9 minus log base 3 of 81. So this property tells us–

This is the same thing as– Well I’m ending up

with a big number. Log base 3 of 1/9

divided by 81. So that’s the same thing

as 1/9 times 1/81. I used two large numbers

for my example, but we’ll move forward. So let’s see. 9 times 8 is 720, right? 9 times– Right. 9 times 8 is 720. So this is 1/729. So this is log base

3 over 1/729. So what– What does– 3 to

what power is equal to 1/9? Well 3 squared is

equal to 9, right? So 3– So we know that if 3

squared is equal to 9, then we know that 3 to the negative

2 is equal to 1/9, right? The negative just inverts it. So this is equal to

negative 2, right? And then minus– 3 to

what power is equal 81? 3 to the third power is 27. So 3 to the fourth power. So we have minus 2 minus 4 is

equal to– Well, we could do it a couple of ways. Minus 2 minus 4 is

equal to minus 6. And now we just have to confirm

that 3 to the minus sixth power is equal to 1/729. So that’s my question. Is 3 to the minus sixth power,

is that equal to 7– 1/729? Well that’s the same thing as

saying 3 to sixth power is equal to 729, because that’s

all the negative exponent does is inverts it. Let’s see. We could multiply that out,

but that should be the case. Because, well, we

could look here. But let’s see. 3 to the third power– This

would be 3 to the third power times 3 to the third power

is equal to 27 times 27. That looks pretty close. You can confirm it with

a calculator if you don’t believe me. Anyway, that’s all the time

I have in this video. In the next video, I’ll

introduce you to the last two logarithm properties. And, if we have time, maybe

I’ll do examples with the leftover time. I’ll see you soon.

## 100 Comments

## GODzbleach

My stupid teacher made it look complicated and got confused but this guy makes it look simple amen to you

## Cameron the Creator

Who has an exam tomorrow??

## Karen Perea

I'm I the only one who see's pink?

## TheHeir ToEarth

I thought you add the exponents with the same base (example 1)?

## Rachel Lim

God, I love you

## Willis96

how does this have dislikes? this channel is better than my high school maths teacher..

## David Dorfman

Awesome video! Upload in HD

## Pratik KAdam

lol

## magua4

wow. these videos will single handedly save my grade in precal. will watch more. very helpful

## MindCrushExodia

in 10 minutes*

## Funny Randomz

Why did you divide A-C in 6:20 when it had minus?

## Gew Was Here

My teacher directly combined all the laws and I was like "What the heck happened?" Thank you Khan 🙂

## surelysavvy

I'm sitting here before my math exam in college going over this. I've struggled with logs since day one with my math professor and you cleared everything in 10 minutes. Thank you SO MUCH! <3

## P S

Its funny how my teacher cannot teach us in 60 minutes but this guy taught it to me in 10. Thanks so much!!

## gerpums

i wonder if you know how much you help people khan. :<

## ClassicalBeast

rentafriend .com …..btw

## Tapan Kumar Basu

Good write up

## Tapan Kumar Basu

Good Write up

## Katagawa Suguieeneh

Thankyou man for this video, really useful. The problem is i TOTALLY know the mathof the powers and its [rp[erties. But i don't have any idea for the logs with base no.s, no idea how to solve them. I read many articles and tutorials on this, almost all is telling more complicated and extra bullshit. Thumbs up bro.

## xlus3

you teach better than 90 percent of teachers today

## AeolusMN

It's wierd, i was taught in my country that when writing a logarithm the base goes left and the power to which the base was raised goes right. So instead of log2(8)=3 i write 2log8=3. I constantly have to think about it when following english examples. That aside thanks for the vids.

## Amera Hassan

I can't even begin to tell you how much I laughed during this video.

Either I have a "different" sense of humor (I do love laffy taffy jokes), or you're just hilarious.

## None of your business

I thought when you multiplied a fraction with a whole number you place a 1 under not over, or am i wrong

## Zachery Tan

Thank you so much I learn the basic of log in just 10 minutes 🙂

## Ryan Post

I love all your videos, this helped me pass college algebra with a B! thanks man!

## Lynchie Mias

tnx 🙂

## Nicholas Anderson

Thank you very much Khan. Appreciate it.

## Nick Bray

240p! Those graphics! Crystal clear!

But really, this helped so much! My math teach would draw these stupid arrows to describe the rules instead of saying, "When x happens, this goes here. When y happens, this goes over here." I didn't really like the arrows, and this is way better. Thanks.

## Solar G

How come you divided with the minus sign and on the next example you multiplied 1/9 . 1/81,

## Amna H.

Because of your videos I'm able to study for finals in 2-3 hours

## Atul Sharma

do you have more videos on logs …I never cared for maths as much as I do now only 20 years later. iam revisiting all modules .. you have done a great great job

## omar saucedo

Excellent videos! Really comprehensive! 😀

## Fernanda Rodriguez

but if you'r dividing 1/9 / 1/81 shouldnt be 81/9 so log3 (9) = 2?

## Eric Tang

Amazing VLikes and susbsr]criened!

## swelihle sphume

hahah thanx but whatsupp with the "aaaah i dunno"

## NNJAx

captured in stunning hd

## Armando Bernal

your penmanship needs serious work here, is this log3(A+C) or (A.C)?

## Jay Lai

Thanks to this vid, I can do a crash studying for tmrw test. Lolols.

## hte spageti

exxplains better than my professor

## morgengabe1

"What do they call this? The additive exponent… I dunno, I don't know the names of things…"

## morgengabe1

Also, 240p looks really good on my phone's screen.

## Nathaniel Mensah

doing a great job khan thanks a lot mate

## Maurice Muteti

thanks khan am learning alot here

## Mert İdil

Where can I found the proof videos ?

## Marnie

this is sucks

## Andrew Maxwell

These videos are really useful, but it would be nice if the older ones were reuploaded in higher quality. If it were 480p or something it wouldn't be much of an issue, but at 240p they're actually difficult to read at some points.

## Math Vault

Cool. Here are 9 properties of log, if it helps: mathvault.ca/logarithm-theory/

## Abraham Hamadi

what property is this called

## Sadia Tahir

is there any sequece of these videos?

## Addhyan

We all are only interested because THIS IS NOT SCHOOL. THIS IS YOUTUBE. Learning about girrafes pooping is interesting on youtube

## Aaron Keith

"Let me do a more cheerful colour…" Picks blood red.

Anyway, great videos!

## Rachel Palacios

Idk who this guy is, but he just saved my freakin life!…the way he explains things is just perfect, it registers in my brain perfectly. and I'm just so pleased to have found this youtube channel.

## Kassandra Fernandez

Casually struggles with homework while I just "learned" this today

## joseph van

gang gang

## thebutter flyNL

4k?

## Christian Aleman

your voice soothes me

## shex bahoz

It's a very good video! If you don't believe me, you can check it yourself

## Dahlia Richards

thanks😃

## Kiran Tech-er

don't teach like u r teaching to a baby

## BCM 2313

Let's use some… hm Idk examples

## MEAT_DUCK117

this 240p xD

## Neven Terzic

I cant understand how did you get 16

## Alecto

I'm so thoroughly amused at a new method to add 3 to 5. Who needs simple addition when we have logarithms!

## David 0221

Umm uhh hmmm the quality tho 😫😫😫😭😭😭

## Natalia Ramirez

I´m gonna pass my finals thanks to these videos

## SavageStyle

WHERE IS LINK TO PROOF? …

## Sameer Shukla

!!! r u kidding me was it this much easy from always .i cant believe this guy really explains well and he keeps it short and smooth ,i loved it just keep helping me like this and changing colours also

## Steven Choi

You sound so much more excited, younger, and enthusiastic in 2007 then 2012 xD Thanks dude 😀

## DiliP Kumar

Indices & logarithms in Hindi basd on jssc

## Mr Mpho _ Mr 101%

your videos are great but please improve the writing method

## Wasique Hussain

Good one for basics

## R A

You sound like DJ Vlad haha

## Ryan

I died a little inside when he said computer science majors should know that 2^8=256 because I didn't know that.

## Byron JONES

thanks mate another great maths video

## Rin

,,sounds kinda like Kim Namjoon

## XephyrCraft

2007?? Where have I been!!!

## Ki-Adi Mundi

I'm in grade 5, wat am i do'n with ma life?

## Bleh _

A true life saver

## J. cali

I laugh every time he says "I DONT KNOW" becuase ME 😂😂😂

## Jasper Fong

Thanks for the good videos dude, keep up the bud work

## Arman Ramezan Nejad

Thanks for video, but the 2nd Log power is 1 over 81, am I right?

## Adhd Millionaire

Quality is killing me redo this video if you can having trouble following along

## Waano Channel

The video was uploaded in 2007 😲😲😲

## Umer Khan

Best teacher

## Red Gears

How do you multiply large numbers in your head

## William Nijsten

How is 81 the same as 1/81? Isn't that 81/1?

## Labyirnth II

I love this man

## Hrishikesh Harathi

great video mate!!

## Una Ex Gachatuber

This vid came out the same year i was born

## Justin Melendez

Thank you, Khan Academy, for teaching me better than Ms Mishra ever could

## Mustafa Ozery

2019

## Riya Fartyal

Set the quality to 240p. Thank me later.

## HnW

more cheerful color: blood red

## Ankur Mishra

It is Modification of Adding exponents

## Manuel Becerra Escudero

2019 Wassup 😀

## Sean Alexander Aguala

1:46 “lock a them” 😂

Loved the tutorial!

## Sean Alexander Aguala

Shouldn’t

2 cubed • 2 to the power of 5 = 4 to the power of eight?

## liffi D monky

where are the videos of teh demostration of the properties?

## Chester Carbungco

why log3^81 = log 4? I'm thinking if that is 64? not 81 correct me if Im wrong tnx

## Jotaro Kujo

5:40 shouldnt it be 4^3+5?