Hi guys, I'm Nancy, and I'm going to show you logarithms.
"Logarithms" sounds a little pretentious
so I'm just gonna call them "logs"
and here are the kinds I'm gonna show you.
So if you see something up here that looks like the kind you're trying to figure out
you can use the links in the video or the description to skip ahead.
So I'm gonna show you first the basic kind of logs
then I'll show you some weirder kinds
if you have a log of a negative number, 0, 1, or a fraction.
I remember hating the fraction kind.
And then I'll show you natural logs
which are a special kind of log.
And finally, some even weirder logs
and by that I mean if you have an x in the log part of your expression
or if you have to use the change-of-base formula.
So let me show you the basic ones.
OK, what if you have a basic log expression like this?
What is that?
How do you even read that expression?
The way you read it is "log, base 3, of 9".
But how do you evaluate it? Or what if you have to simplify that?
How do you do it?
There are a few ways.
One way, if you're the kind of person who loves to do things in your head
you can just think to yourself
"3, raised to what power, gives me 9?"
And if you can do it that way, more power to you, you are cooler than the rest of us.
But for all the rest of us
usually the easiest way
the fastest way is to re-write this into exponential form.
And I will show you how to do that.
But the first thing you should do
is if this doesn't already equal something, make it equal to x.
And there's a reason for that.
Because you're going to rearrange this and connect it to an exponential version.
How do you do that? There's a pattern you can use.
And here's the pattern.
To re-write this into exponential form so that you can find the answer
you start with this little base that I circled.
This is the base of your log expression.
Start there, and move in this direction of the arrow...
and raise that base to the power of what's on the right side.
So the base 3...
raised to the power of the right side, which is x...
equals...set it equals...
9, the middle number, or the other number that's left that you haven't used.
Alright, so I've just rearranged into exponential form with that pattern.
Some people think of it as "little, to the right, equals middle"
if that helps you
but that's the general way that you write this.
That's the order
And you end up with something like this that's exponential.
3 to the x power equals 9
which is great, 'cause all you have to do is figure out what x makes that true
and since...
since 3 times 3 equals 9, or 3 squared equals 9...
then you can tell that x must be 2.
And since x equals 2, that's your answer for the log.
That log is equal to 2.
The log is always equal to a power, from the exponential version.
Alright, so your answer's 2.
If it helps, you can think of this order as a snail.
Sounds dumb, but for some people it makes this stick, the order.
The idea is that the circle part, the head of the snail, is your base
and then you move in a spiraling direction kinda the way a snail shell spirals
so counter-clockwise spiraling
and if you do that, you'll get the right order of...
base, to the right side, equals the middle number.
Doesn't really look like a snail, maybe if I added...
more spirals maybe, pretty rough.
So if that helps you, great
but let me show you another log expression.
Alright, what if you have a log expression like this
log of 10,000 and there's no little base written here?
I'm showing you this one, because if you don't have a little base given to you...
it's going to be 10. The default is 10.
It's implied, and I think it's good to go ahead and write it.
So this is actually log, base 10, of 10,000.
Say that you need to simplify this or evaluate this...
I think the best way is still to go ahead and write "= x"...
and to connect it to the exponential version of it.
And that will let you solve for what the log equals.
So remember, you start with the base number, the small base number here...
you raise it to the power of what's on the right side, which is x...
and then you set it equal to the other number
the number that's left, the one in the middle, 10,000.
OK, so you have an exponential version of it now.
All you need to do is figure out what x makes this true...
what x makes 10, raised to that power, equal to 10,000...
and since 10 times 10 times 10 times 10 equals 10,000...
or 10 to the 4th power equals 10,000...
you can tell that that little power x has to be 4
because 10^4 equals 10,000, so x equals 4.
And now because x equals 4
your whole log expression is just equal to 4, and that is the answer.
Alright, here are some weirder kinds of log I want to show you.
Log of a fraction, log of 1, log of 0, log of a negative number...
You could see some like this.
What if you have a log of a fraction?
This is log, base 2, of one-eighth.
How do you find that?
For these, still do the same steps as before.
Set it equal to x, rewrite it into exponential form...
so when you re-write this, you get 2^x = 1/8 over here.
All you need to do is find what x makes this true
but it's not obvious when you have a fraction like 1/8 on the right side.
You don't want that. You want to somehow make this into 2 to a power.
There are tricks for this.
First, check to see if anything in this fraction is a power of 2...
8 is 2 to the 3rd power
so one trick is to re-write this as 1 over 2^3 instead of 1 over 8.
And then the other thing you'll need is to re-write this.
1 over 2^3 is the same as 2^(-3).
As you probably know...
2 to the negative 3 power
the negative power just means 1 over 2 to the positive version of the power.
So when you do that...
all you have to do is compare x and this power, the negative 3
to see that x is negative 3...
and since x is -3, this whole log is equal to -3...
so the log, base 2, of a fraction is probably a negative number, a negative power.
What about log of 1?
This looks so simple, and yet it can be very confusing...
confusingly simple.
First of all, if there's no little power there, remember that it's 10.
It's a hidden 10, and it's probably better to write it.
Set it equal to x, because you don't know what it is.
Just set it equal to a variable, re-write it...
in exponential form, this is 10 to the x power equals 1.
So what power, when you raise 10 to that power, equals 1?
Well, if you didn't know...
any number raised to the 0 power equals 1.
So 10 to the 0 power equals 1.
That's the only way that will happen.
So that whole log is just equal to 0.
What about log of 0?
Same idea.
For all of these, just try to re-write them and see what happens.
And when we re-write it, we get 10^x = 0.
If you think about this...
there's never any number you could put in for the power of 10...
that would ever give you back 0.
Every power you put there...
will give you a positive number greater than 0.
10 to the 0 power is 1, not 0.
10 to a negative power is not a negative number.
It's some fraction like this.
So there's no x that will ever make this work.
So x is undefined.
This log is undefined.
And finally, log of a negative number...
same steps, try them...
and when you do try that, you have 10^x = -1
Just like in this one, you will never get negative 1 back.
It's impossible.
10^0 is 1.
10 to the negative number is some fraction that's positive.
This is impossible
so x is undefined. There's nothing that works.
The log is undefined.
So these were my weird examples of logs. Fractions, negative numbers, etc...
Now I wanna show you something called the natural log.
Alright, now let me show you natural logs.
That's what this ln means. It stands for natural log, so natural log of 1.
Or ln of 1.
What is a natural log?
It really is not as hard as it sounds.
It's just a special king of log
where the base is e.
So whenever you see ln...
you can re-write it as log, base e.
I think that's the easiest way to evaluate it.
So ln 1, natural log of 1, is really just log, base e, of 1.
So if you want, you can re-write it
and then it'll be easier to solve for what this equals.
What is e, by the way? It's just a special number in math.
It's a constant, and if you put it in your calculator
it'll be some decimal like 2.718, something, something...
and so on, forever...
but it really is just defined to be the base of the natural log.
So you don't really need to understand that.
It's just that, when you see ln...
just know that it means log, base e.
After doing that
all the other steps are the same as what you've been doing before, so...
set this equal to x. Rearrange it into exponential form.
So if you rearrange this into exponential form
you start with the base, which happens to be this e symbol...
you raise it to the x power, and set it equal to 1.
So e to the x power equals 1.
And after that, just ask yourself
"What power would ever make this equal to 1?"
And the only one that works is 0.
Remember, a number raised to the 0 power is 1, always
so x has to be 0...
and since x is 0, it also means that ln of 1 is also 0.
Here's one more natural log problem.
ln(e^3), so natural log of e to the third power
or natural log of e^3.
This looks complicated.
I promise it's going to be very easy for you.
I still think the best way to do this
is to re-write the ln as log, base e, so let's do that.
Looks a little strange, but that's what that stands for.
From here, just use the same steps.
So how did I get this?
Same steps as before.
Take the base e, raise it to the right side power, x...
and set it equal to whatever was in the middle of the log
in the argument of the log, which happened to be e^3.
So whatever that was, that's what you set it equal to.
And after you do that
all you need to do is match theses, compare them
and see that x has to be 3.
And since x has to be 3, this whole log is just equal to 3.
So that whole expression was just equal to 3.
Now there's one more kind of log I want to show you
and that is when you have an x somewhere in the log expression
in the base or in the argument.
So it's a little weirder, but it's the last type, I promise.
So let's look at those.
Alright, here are some even weirder types I wanna show you.
They're even weirder, because...
they might throw you off if you see them.
They're a little more confusing, because like in these two...
the x is inside the log expression.
It's the base here.
And it's the argument, the middle number, here.
And then this last one is super weird.
It will mean that you have to use the change-of-base formula
which you might have heard of.
But let's look at these two.
If you see one that says "solve"
and it's a log equation with x inside the log...
I know it looks different, and...
solving log equations is actually a whole other topic
that could take a whole other video, but I wanted to give you a taste of it
because I know it looks a lot like what we were doing
and I don't want you to be thrown off if you see it.
You really can use the same steps as before.
So try it. For both of these...
if you see something like this
even though there's a variable inside the log
you really should try the same steps
because you'll probably get something that's a lot easier to figure out.
So let's try it for the first one.
So when you rearrange it, you take the base, x...
raise it to the 5th power, on the right...
and set it equal to 32.
x^5 = 32
And then looking at that, you can tell that
x has to be 2
because 2 to the 5th power equals 32.
And that really is the answer.
x equals 2.
So just try the same steps as before
and you may be able to figure out what the solution is.
Same thing for this one. Rearrange it...
so you get base 5, raised to the power of 3, equals x
because x was the middle number.
And this is even faster.
You just need to figure out what 5^3 is.
5 times 5 times 5 is 125...
so the answer is x equals 125.
So if you see one like that, just try the same steps, that's...
that's the bottom line.
And then this last type...
calculate log, base 2, of 7...
log, base 2, of 7, I mean that looks a lot like what we were doing before
and you should try to rearrange it the same way.
So if you do...
if you do, you get 2^x = 7
and I can't think of a number that makes that true.
I don't know what that number is.
If this were 8, it would be 2 to the 3rd power...
but there's no nice, neat integer number that works.
It's going to be a decimal that's hard to find by hand.
So what you have to do, if you run into that
is use the change-of-base formula...
and whatever log you had, log, base 2, of 7...
the change-of-base formula lets you re-write it.
So you can re-write it, and it becomes
log of 7, the larger number, over log of 2, the base number.
Log of 7 over log of 2
and you can use whatever base you want.
This is just like the default 10
but you plug this into a calculator, and you do that division
and whatever decimal you get is the answer for the log.
I'm showing you that in case you get one like that
where there aren't these like nice, neat numbers that you can do by hand
so that's important, in case that comes up.
So I hope that helped you understand logs.
I know logs are everyone's favorite.
It's OK, you don't have to like math
but you can like my video
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