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>>In this video we define
the common log, log of x
and the natural log,
the line of x
and we do the following
8 problems.
This is part 11 of logarithms.
We're going to begin
with talking
about what a common log
and a natural log is.
So the common logarithm
is base 10.
What I mean by that, if
we just write log of x
and you notice I don't
write anything for the base,
this is called the common log
and that just means
log of x base 10.
So it's similar to using
this square root symbol.
Do you know when you put the
cube root of, let's say 64,
that tells you what
number cubed to 64 is 4.
But if you just put
the symbol like this,
it means the square root of 64.
We don't usually write
the little 2 in there.
So it's the same thing.
When we use this radical symbol
and we don't write a
little number in there,
it means the square root.
It's the same thing
with the logs.
When you write a log and
you don't write the base,
it always means base 10.
And that's because we have
a base 10 system, basically.
So, if I say what is the log
of 100, I want to know what
that equals, you have to
know that the base is 10.
And then you would think 10
to the what power equals 100?
Can you figure that
out in your head?
That'll be 2.
Now, of course, if you couldn't
think of it in your head,
you can go back to
writing 100 as 10 squared
and then use your rules
as the space is the same
as that, you get a 2.
In other words, 10
squared equals 10 squared.
Let's do another 1 like that.
How about log of 0.001?
What's that going to equal?
That's little bit trickier.
Remember, the base, even
though you don't see it, is 10.
So the question is 10 to
the what power equals 0.001?
Do you know what that would be?
That would be negative 3.
Now, if you're not sure how
I got that, you can fool
around with 0.001, if you
don't know how to write
that in scientific notation.
But in scientific notation
it is 0.001 is written
as 10 to the negative.
So that's 1 way of doing it.
And to figure that out,
you still have a little bit
of trouble with it,
you could write
that as 0.001 means
1 thousandth.
What's 1 thousandth?
That's 1 over 10 cubed.
I'm going to write this
10 a little bigger.
And then, I could use my
laws of exponents to write it
as 10 to the negative 3.
In any case, it's negative
3 and you should know
that when you see a 1 with
0s in it with a decimal here,
you should be able to write it
in scientific notation
and figure that out.
But if not, as you can
see, there are other ways
to eventually get
the same answer.
Well the ones I gave you, it
was easy, you could figure
out what the exponent
of 10 would be.
But let's look at this.
What's the log of 13?
10 to the what power equals 13?
Well, that's a little
bit tricky.
You know it's not 1 because
10 to the first would be 10.
And if were 2, that
would be 100.
10 squared would be 100.
So it's between 1 and 2.
So you could estimate
that this is going
to be 1 point something.
But the question is, how
would you know what that is?
Well, we could use a calculator.
Now, all calculators are
a little bit different.
So it depends on if you're
using a graphing calculator
or a scientific calculator.
So in a scientific calculator,
usually there's 1
or 2 ways to do it.
You would enter the
number first,
then you would punch
the log button.
And if that doesn't
work, you might also need
to press the word enter.
In mine, I don't
need to do that.
In my calculator, if I push 13
and then you should
see a log button,
it comes up automatically
that I'm going to get 1 point.
I'm going to write it up here.
1.1139433352.
And that's not the exact
answer but that's as far
as my calculator goes.
So really, this goes
on and on and on.
So if we were going
to approximate this
to 4 decimal places, which is
where we usually estimate
logs, 4 decimal places.
Let's see.
If I rounded this,
this would be 394ths,
since I cut it off there.
4, you're just going
to keep it the same.
So that's 3 9 and so it's
approximately 1.1139.
So try this 1 in
your calculator.
How about the log of
6 See what you get.
In my calculator,
I get 0.77815125.
That really goes
on and on and on.
And if I were going to,
again, 4 decimal places,
this would be 0.77- okay, I'm
cutting it off after the 1 here.
And it's a 5, so
that's going to be 8 2.
That would be rounded
to 4 decimal places.
Does that make sense?
We're saying does 10 to the
point 77 blah, blah, blah.
Does that equal 6?
That's what really
you're saying,
remember the log is an exponent.
The log of 6 is the
exponent, 9 10 it equals 6.
Well, 10 to the first
power would be 10
and I only wanted to get 6.
So it makes sense that
this is less than 1.
So that's why it came up
0.778, etc. So you can estimate
and make sure that
it's reasonable.
At least you know it's
in between 0 and 1.
The other special log is log
base e. First of all, what's e?
You might have heard
of it before.
E is an irrational
number, like pi.
They are very useful in math
and you will see some uses soon.
It's also very important
in calculus.
E is 2.71828182...It does not
repeat, even though you happen
to see 1 8 28 twice in a row.
The next digits don't repeat.
So it's approximately 2.7.
Just like we memorized pi
to be approximately 3.14.
So use this number,
it's about 2.7
and the way we write
the natural log is ln.
So if you put ln
of, let's say, 10.
That means log base e of 10.
So just like the common
log, we don't write the 10;
for the natural log,
we don't write the e.
So if you see log base e, the
shortcut is to write ln. Now,
in books, that usually
isn't in script.
I like writing in script
so it doesn't look
like the word, "In."
Again, we can use
our calculator.
So let's say we wanted to
get the natural log of 10.
So on my calculator, I'm going
to put the 10 and the ln button,
and you might have to put enter.
That's- you might have
to use that button.
So let's see.
I'm going to put in 10.
And I press ln and it
comes out 2.302585093.
And, of course, remember
that goes on and on and on.
And let's round that
to 4 decimal places.
What will that be?
2.3026. Now, let's see
if that makes sense.
Log base e of 10 is
approximately 2.3026.
So what you're wondering
if that makes sense.
Do you remember e is
about 2.7, that's about 3.
Imagine e's not quite 3.
So kind of like 3 squared is 10.
Yes, it seems in the ballpark.
Of course, hopefully it's right
because hopefully your
calculator is computing
it correctly.
So, this is a natural log.
Try 1 more.
How about the natural
log of, let's say, 213.
And how about approximating
it to 4 decimal places.
So go ahead and try
that on your calculator.
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>>All right.
I got 5.36 and it goes 1, 2,
9 so it's going to be 1 3.
Okay. So that's about the
common log and the natural log.
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>>Here are 2 quick
problems for you to try.
All right.
Natural log of e to the 5th.
Remember, you don't
have to write this out,
but it means log base
e of e to the 5th.
So, really, it's just
the exponent here that's
in your laws of logs.
And same thing here.
This means log base e. Now
what square to e means.
Either the , so the
answer is just .
So if you've got the natural
log, and you've the base e,
then it's the exponent
of the answer.
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