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## Get ready for AP® Statistics

### Course: Get ready for AP® Statistics>Unit 3

Lesson 3: Logarithms

# 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?
• It's actually written "ln" instead of "nl" because the Latin name of natural log is "logarithmus naturali."
• 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?
• 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.
• If I am looking for a calculator to help me with this should I get a scientific calculator or graphing calculator?
• 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.
• Where does "e" come up in nature? Just curious.
• "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).
• 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?
• 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. :)
• What is the difference b/w log and ln??
• 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.
• Where exactly is the number "e" found in nature? Why do people call it a natural number?
• 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.
• why is a logarithm with base e called the natural logarithm? What makes it "natural"?
• 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.
• Does anyone know how to change the base on a TI-83 to something other than ten?
• Ryan Farias is right TI-83 cannot change the base without doing a different formula. I am confused as to what that formula is though... I know that it has something to do with dividing the base and/or the log by one or the other.
• What is a natural log used for? Is there an explicit reason they chose
``e``
?