# Introduction to logarithm properties | Logarithms | Algebra II | Khan Academy

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.