Welcome, ladies and gentlemen.
So what I'd like to do is show you
how to evaluate some logarithmic expressions when
we're just kind of using the properties of logarithms.
And what I did here is-- actually,
let me step out of the camera here real quick.
What I did is I highlighted some very common
logarithmic properties that we're
going to be using to help us evaluate
these logarithmic expressions.
Now, some of these you might be familiar with.
Some of these might be a little bit new.
But using these properties is going to greatly help
us evaluate logarithms because what we're going to do
or what you've previously done is
learn how to evaluate a logarithm for what exactly
a logarithm means.
Actually, you should already know that.
So for instance, if I said, let's use the basic one.
If I said log base 2 of 4, hopefully you
understand that the answer to this is going to be 2,
and the reason being is because what the question is asking
is 2 raised to what number gives you 4?
Well, we know the answer is going to be 2.
Well, again, if I write 2 raised to 2 equals,
what does that equal?
So again, I'm asking 2 raised to what number is equal to 2?
Well, hopefully we understand that that number
is going to be 1.
Log base b of b is just equal to 1.
Now, there's another thing you guys can think about.
Any number you can always raised to the first power.
So log base 2 of 2 to the first power,
you could also think of that as equaling 1 from there.
So let's go back to a step further.
Let's go back.
And what about if I wrote this as 2
raised to the fourth power?
Now let's change this to a 3.
Let's see what this would be.
Now, log base 2 of-- oops, now, these aren't same powers,
so that doesn't follow that rule, does it?
So let's change this to 2.
And now let's raise this to the third power.
So 2 base 2 of 2 raised to the third power.
Well, by using my previous logic of 2 base 2 of 2
equals 1 and up to 1, then therefore we
could probably say that this is going to equal 3.
Now let's go ahead and verify our answer here.
If I had 2 cubed, that's 8.
So log base 2 of 8-- that's really the same thing.
Well, 2 raised to what number gives you 8?
Well, 2 times 2 is 4.
2 times 2 times 2 is 8, which is 3.
So therefore, this you can see how this rule
is going to apply, as well.
So log base b of b raised to the power
is going to just be equal to whatever that power.
So the last property we're just going to look at
is basically kind of like the exact [INAUDIBLE] this.
If I have the base b and that's raised to my log
base b of x, therefore then that's just going to equal x.
And again, we can just look at this.
I'll just use some numbers as a property here.
So let's use 2.
Let's do log base 2 of-- again, let's do 4.
Or let's do 8 again.
That's fine.
Let's use 8.
Now, we know 2 raised to what power gives you 8?
That's 3.
Or log base 2 of 8 is 3 because 2
raised to what power give you 8?
That's 3.
So in reality, that's going to be 2
to the third power, which is 8.
Or based off of that logarithm, based
on that rule, when we have 2 raised to the log base of 2,
we know that the answer's going to be 8.
So you could either look at the best thing
to do is I wanted to spend some time since these are going
to not take that long, these are not
going to take-- I wanted to spend some time explaining you
and showing you how these rules work.
Now, let's get into some of these examples.
So for the first example, we have log of 10.
Now, make sure when we see log of 10,
we understand that that is log base 10 of 10.
So therefore, I'm taking the log.
I have the same base as what I'm evaluating the logarithm for.
So therefore, by using this rule,
I know that that is going to evaluate to equal 1.
Then we introduce the natural logarithm, which is e.
Is that the only one I did?
I did, yeah, OK.
So when we have a natural log of e, remember, that base is e.
So I have ln e of e.
That, again, is going to be another example of 1.
So we have the log, natural log.
And then for logarithm, I could choose really any base.
So if I have log base 3 of 3, we know that answer is, again,
going to equal 1.
Again, the first three problems, all I did was use that rule.
For the next two or next four, I'm
going to be using this rule.
So again, you can see here, this basically
is the exact same thing.
And even sometimes my 6's look like my b's.
So we have log base 6 of 6 raised to the x.
Well, like I showed you in that previous example,
my simplified answer is just going to be x.
Over here, we have a base e.
So it's ln base e of e.
So therefore, we know we can just
evaluate that as negative 4.
Now, in this case, we can get a little confusing here
because over here, we have log base 2 of 32 equals x.
Well, 2 and 32 are not the same.
However, if I can rewrite 32 as a base of 2,
then they are the same.
Well, let's think about it.
2 squared is 4.
2 cubed is 8.
2 to the fourth is 16.
And 2 to the fifth is 32.
So guess what?
I can rewrite 32 as log base 2 of 2
to the fifth power raised to the x.
Then by using my power rule, I can multiply the x and the 5.
So I have log base 2 of 2 to the 5x.
Now you can see now I use my rule.
And the log base 2 of 2 is just going to give me
a final answer of 5x.
So exact same thing happens here, i.e.
[? 3 ?] and 81 are not the same.
However, if you start writing out 3 raised to a power,
you have 9.
3 cubed is 27.
3 to the fourth power is 81.
So therefore, I can write this as log base 3 of 3
to the fourth raised to the negative x.
Then, again, simplifying this, I have log base 3 of 3
to the negative 4x.
Well, now, again, the log base 3 of 3 follows this rule.
There is going to divide out.
So therefore, you're just left with a negative 4x.
Hopefully that still shows up on the video, negative 4x
over there.
Over here, so now the last examples,
we're just going to be using this rule, which
I explained last.
So if I have e base to the ln of e,
just remember that e is in that base.
So e raised to the ln of b, that's just going to equal 4.
Over here, I have 7 raised to the log base of 7x.
That's just going to equal x.
And the last example, I wanted to have a little fun
because I know usually people will look at this and be like,
oh god, look at that-- crazy.
I don't know how to do this.
But again, if you just follow the rules,
10 raised to the log base of 10-- 10
raised to the log base of 10 just
equals whatever you're evaluating for.
So therefore, my answer is just 7x squared minus 3x plus 1.
There you go, ladies and gentlemen.
That is how you evaluate logarithms
by using the properties of logs.
Thanks.