# Evaluating natural logarithm with calculator

## Video transcript

Use a calculator to
find log base e of 67 to the nearest thousandth. So just as a reminder, e is
one of these crazy numbers that shows up in nature, in
finance, and all these things, and it's approximately
equal to 2.71 and it just keeps
going on and on and on. So you could view
log base e as 67. Let's see, what does e
mean? e is just a number, just like pi is just a number. So this is really the same
thing as saying log base 2.71, and the actual
numbers, so you'd have to write all the digits
that keep on going forever and never repeat 6 of 67. So what power do I have to
raise e to to get to 67? So another way of saying
that is this is equal to x. You're saying e to
the x is equal to 67, we need to figure out what x is. Now, traditionally
you will never see someone write log
base e even though e is one of the most common
bases to take a logarithm of. And so the reason why you
wouldn't see log base e written this way is log
base e is referred to as the natural logarithm. And I think that's used because
e shows up so many times in nature. So log base e of 67, another
way of saying that-- or seeing that, and the more
typical way of seeing that is the natural log. And I think this
is ln, so I think it's maybe from French or
something, log natural, of 67. So this is the same thing
as log base e of 67. This is saying the
exact same thing. To what power do I have
to raise e to to get 67? When you see this ln, it
literally means log base e. Now, they let us
use a calculator, and that's good because I don't
know off the top of my head what power I have to raise
2.71 so on and so forth-- what power I have to raise
that to to get to 67. So we'll get our calculator out. So we get the TI85 out. And different
calculators will have different ways of doing it. If you have a graphing
calculator like this, you literally can literally type
in the statement natural log of 67 then evaluate it. So here this is
the button for ln, means natural log,
log natural, maybe. ln of 67, and then
you press Enter, and it'll give you the answer. If you don't have a
graphing calculator, you might have to press 67
and then press natural log to give you the answer,
but a graphing calculator can literally type it in the
way that you would write it out, and then you would press Enter. So 4.20469 and we want to round
to the nearest thousandth. So this is the thousandths
place right here, this 4. The digit after that is
5 or larger, it's a 6, so we're going to round up. So this is 4.205. So this is approximately
equal to 4.205. And it actually
makes a lot of sense, because we know that
e is greater than 2, and it is less than 3. And if you think about
what 2 to the fourth power gets you to 16. And 3 to the fourth
power gets you to 81. 67 is between 16 and 81
and e is between 2 and 3. So at least it
feels right that's something that's like
2.71 to the little over the fourth power should
get you to a number that's pretty close to 3
to the fourth power. Actually that makes sense
because it's actually closer to 3. 2.71 is closer to
3 than it is to 2. So this feels right, that
you take this to the fourth, little over the fourth
power, you get to 67.