If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

### Course: Algebra 2>Unit 8

Lesson 2: The constant e and the natural logarithm

# Evaluating natural logarithm with calculator

Sal evaluates log_e(67) (which is more commonly written as ln(67) ) using a calculator. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• So does anyone know if he was right about that "log natural" french thing?
(86 votes)
• It's actually written "ln" instead of "nl" because the Latin name of natural log is "logarithmus naturali."
(270 votes)
• When Sal says e shows up in nature a lot, what does he mean? Where in nature does e show? Like, I know pi, in nature, is the ratio of circumference to diameter, is there any such thing for e?
(24 votes)
• In my work, I encountered e a lot more than π. The constant shows up in exponential functions all the time, such as in radioactive decay. It even shows up in such things as statistics, business math, civil engineering, and computing interest -- just to name a few.

As far as why we would use such a strange number as e as the preferred base for a logarithm, that will become very evident if you go on to study calculus. For now, let us just say that the math is so much easier with the natural logarithm that in higher levels of math and many applied uses of math, the natural log is used almost to the exclusion of any other base. There are some specialized fields where it makes more sense to use a base 2 or base 10 logarithm, but the natural log is far, far easier in the vast majority of applications.
(49 votes)
• If I am looking for a calculator to help me with this should I get a scientific calculator or graphing calculator?
(12 votes)
• The TI series of graphing calculators are an excellent brand of calculators. I personally use the TI-83plus, because it is cheaper than the TI-84 and not a whole lot different than the newer version (besides being able to change the base of a log. If that's important to you, than consider the TI-84plus, but its more \$\$\$). As for scientific calculators, you should definitely have at least one because they are so much easier to operate than graphing calcs. Texas Instruments sells a variety of these simpler (but very useful) devices. For a final note, I wouldn't buy more than one graphing calculator -- often computers can do the same things graphing calculators can, but even faster. Because of this, graphing calculators are more for students, whereas adults who need to do that sort of stuff usually use their laptops for the job. This reply might have come a bit late for jtfeliz, but I hope it shines some light on selecting a calculator that's right for you for anyone else who needs a calculator.
(20 votes)
• Where does "e" come up in nature? Just curious.
(13 votes)
• "e" is the natural representation for any problem involving exponential growth. For example, half-life problems are typically expressed at the college level using "e", as it gives you a clean connection between the amount of the radioactive substance remaining and the current rate of decay (the level of radiation).
(20 votes)
• What is a natural log used for? Is there an explicit reason they chose
``e``
?
(9 votes)
• e is incredibly useful in a lot of processes like modelling population growth, radioactive decay, compound interest etc. Due to this, it is commonly used as a log base. As to why it's called the "natural log" is up for debate, but I've seen two reasons:

1. It can explain a lot of natural processes like population growth

2. The derivative (an operation in calculus that deals with the rate of change over very small intevals) of the function e^x is e^x, which makes it unique as it is the only function to have this property (aside from 0)
(16 votes)
• why is a logarithm with base e called the natural logarithm? What makes it "natural"?
(9 votes)
• If you graph the function e^x, then draw the tangent line to the curve at the point (x, e^x), the slope of that line will be exactly e^x. This isn't true for exponentials of other bases.

This fact becomes very convenient and pretty in calculus, so e^x is in a sense the most natural base for an exponential function.

Then ln(x) is the inverse of e^x, so it gets called the natural logarithm.
(12 votes)
• What if you have a number in front of the e instead of log? Like...4e^x = 10? How would you go about doing that?
(6 votes)
• For problems that add/subtract to/from the x, simply solve for the exponent by using ln. In the example you gave:
e^(x-4) = 2
x - 4 = ln(2)
x = ln(2) + 4
An example for division:
e^(x/5) = 2
Same thing as before.
Use the ln.
x/5 = ln(2)
x = 5 ln(2)
For your last example let's equate it to some constant just for the sake of clarity. We'll choose 2 because it's a really friendly number:
e^(ln(5x)) = 2
Now, we have an important identity for logs. x^(log(N) base x) = N
so e^(ln(5x)) = 5x
so 5x = 2 and finally x = 2/5 Hope this helps. :)
(8 votes)
• What is the difference b/w log and ln??
(4 votes)
• ln is the natural logarithm. It is log to the base of e.
e is an irrational and transcendental number the first few digit of which are:
2.718281828459...
In higher mathematics the natural logarithm is the log that is usually used. The log on your calculator is the common log, which is log base 10.
(10 votes)
• So will you always round to the thousandth?
(6 votes)
• You don't necessarily always round to the thousandths, although it is quite common. Every question should tell you what place to round to (this goes for almost all math problems).
(6 votes)
• Where exactly is the number "e" found in nature? Why do people call it a natural number?
(6 votes)
• e comes up all the time in real-world math. For example, it is used in business math for certain kinds of interest calculations. It is used in calculating certain kinds of reaction rates, especially radioactive decay. It is used extensively in engineering computations.
(7 votes)

## 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.