[INSTRUCTOR] All right, logarithms.
Logarithms, why we need them is exactly this right
here, what we're gonna see.
If we're trying to solve for x
And we have 2 to some power is 16, I can think,
okay, well, 2 times what?
2 times itself how many times?
Well, 2 times itself four times is 16,
and so x has gotta be equal to 4.
That one's not bad when it works out equally.
However, 2 to what power is 20?
Well, we just said 2 to the fourth was 16,
and we said we know that 2 to the fifth then is times
another 2, which is 32.
And so 20 falls someplace in between there.
And so we know that x is someplace in between 4 and 5.
But how exact can we get without saying, okay, well,
2 to the 4.2, 2 to the 4.3, 4.4, and doing that,
we need logarithms.
Same deal with this one.
We know that 2 to the fifth power, like we just said,
was 32, and we know that 2 to the sixth power is times
another 2, which is 64.
And so we know that 2 to the 5 point something,
that x is in between 5 and 6 for this one.
But what exactly we need logarithms for?
So, let's explore the logarithm button a little bit here.
So 10 to the 0, anything to the 0 power
is 1. 10 to the first, 10 times itself.
That 10 squared, 100.
10 cubed, 10 times 10 times 10, so we've got 1,000.
And 10 to the fourth is 1, 2, 3, 4, we can just
keep adding those zeros.
So, we've got this log button, fancy log button down here.
And so let's take the log of 1.
Log of 1 is 0.
Log of 10.
Log of 100.
Log of 1,000.
And you can almost guess what the log of
10,000 is going to be.
So log of 1 is 0, 1, 2, 3, 4.
So what do you think the log button does?
Well, what I want us to draw our attention to is this
0 is the same as this 0.
This one, that one, 2 and 2, 3 and 3, 4, and 4.
So, what do we think the log button does?
Well, if we take 10 to the 0 we get 1,
and if we take the log of 1 it gives us
that exponent again.
And so the log is undoing, it undoes
the exponent.
They are inverse functions
of exponents, or exponentials.
So here's what I mean.
If you have
b to the x equals a, we can write that,
and this is called exponential form.
We can write that as log form, saying the log,
which is just another operation.
Base b
of a equals x.
And so this b, we call the base of the log.
And this is what we're taking the log of f.
And so the base of the log and the base of the exponent
are the exact same thing, and then the x and the a swap sides.
If you guys can know this and go back and forth from this form,
you're going to go extremely far with this logarithm concept.
To know that they are the opposite of
each other like that.
And so what we're gonna do is just practice
rewriting this like this.
So, rewrite these as logs.
10 to the third is a 1,000.
So, the log base 10, because the base of the
exponent becomes the base of the log, of 1,000 equals 3.
The log base 5, the base of the
exponent of 625 equals 4.
Log base 2 of 1,024 equals 10.
And so now we wanna find x, and so this is what we were
talking about before.
Log base 2 of 16 equals x. Base of the exponent becomes
the base of the log, and now the x is all by itself.
So if only we could evaluate that.
Now, we didn't really need logs for that,
and our calculator can't do that outright,
but I'll show you how we can adjust it for it.
Log base 2 of 20 equals x, base of the log,
base of the exponent, log base 2 of 50 equals x.
Let's get really good at changing back and forth
between those two things.
So, something we need to know, the common log.
Call it the common log and that's what the log button
on our calculator is, because notice the log
button on your calculator doesn't have a number or
a base, it's because it automatically does log base 10.
Because a ton of our numbers are in the base 10 system,
we work in the base 10 system, and so that's why it's on there.
And the natural log, we already talked about
the natural number being e.
And so the natural log is any log that has the base e.
And so, in both cases, we don't write the bases,
and it's a little easier to recognize, but you need
to notice what the base is.
So what if it doesn't have base 10 or base e,
and we can't use the fancy buttons on the calculator,
log and natural log?
Well, we use the change of base formula.
And so we can change any base, a, into base 10.
And we do the log of x divided by the log of a.
Or you could use the natural log if you wanted
to, whichever your preference.
They give you the exact same answer.
Natural log of x, natural log of a.
And the whole reason behind it is because
you can really change it to any base you want to.
Log of x divided by log of a. This could be base
b and base b, as long as they're the same base.
But most often, we use log base b.
So let's utilize this.
Let's see what we can get.
Let's go back to our other page and say
log base 16, log of 16 divided by log of 2.
And that's 4.
And so that's what we got before.
Log 20 divided by log 2.
We said this one was in between 4 and 5,
and so that one's approximately 4.322.
So that if we go back to why we were saying that, 2 to the
power of 4.322 is about 20.
And then finally, log base 2 of 50.
Let's do the natural log this time.
Natural log of 50 divided by the natural log of 2.
And just to show you
that we get the exact same thing,
no matter which way we do it.
5.644.
All right, we'll come back and talk more logarithms.