In the prior discussion, you looked at exponentials and logs using the common base 10. In this section, you will look at a base that is more useful in the study of calculus; the base e. An irrational number, you can obtain e by solving the following limit (you will look at the definition of limit in the next chapter):
Although you could estimate these numbers by using your calculator and substituting the appropriate values for x into the arguments, a much better way is to use LiveMath's Table feature. Use the second expression above for this example. You may want to try the first expression to verify your answer.Input the following expression.
Select the whole expression and choose Generate Table... from the Compute menu. When the dialog box opens, use the following values.
As you can be see, there are not enough points to give a good approximation for e. You may want to try the same procedure with 1000 as the upper value for x. Try others even higher until you get a value accurate to 7 or 8 digits. How high does x have to be before the approximation is accurate to within 10 decimal places?
Keep in mind that LiveMath generates thousands of points which may require you to allot more memory to the program.
You can calculate the value for e in LiveMath by typing the letter e into a new Prop and selecting Calculate from the Compute menu. Note
For those more familiar with LiveMath, the following is also a method for calculating the number e. The Summation Op shown is discussed in a later chapter.
The Natural Exponential Function, , has as its inverse, the Natural Logarithmic Function. You obtain it by solving for x as you did above for the regular log function. The difference in the answers is basically in the notation. To differentiate this from the log to the base 10, it is labeled with "ln" which is the same as . It is pronounced. "line-x" or "lynn-x". Using LiveMath again, solve for x.
You may want to go back to the example on the common log and assign b = e to see the graph of these two functions.
If you reverse the above manipulation you will achieve a curious answer which may need some explanation. Below is presented what may happen in LiveMath if the process is reversed and you solve for x in the last Prop.
The result is the inverse of the Natural Log Function, as expected, but the notation is different. You can show that this function, called , is really an exponential function, by using the third property of Natural Logs where the following holds true:
You start by stating that ln(e) = 1. If you use the property shown above, you can develop the following equation:
Next, you "exponentiate" both sides of the equation to get the following. To exponentiate means; make an exponent of each expression.
Since the natural log function is the inverse of the exponential function, the left side of the equation simplifies to .
In LiveMath, all you need to do is simplify the expression to see the equivalence:
Above, when the term "exponentiate" was introduced, you were introduced to a method you will find very useful when studying calculus.
Another method that demonstrates the relationship above is to use exponent notation as well as using the exponential function. The equation above could have been written as:
LiveMath will transform,
in most of your manipulations with a Simplify transformation rule. You can see this rule inside the Declarations Prop.
The full list of the Properties of the Natural Log Function is similar to those of the regular Log Function and are given to you below for reference:
Among other uses, these concepts are frequently used in science to describe the growth and decay of radioactive isotopes and population growth phenomenon. In economics they are used to study compound interest, depreciation and effective rates.