lecture

APM 153 LECTURE TWENTY-FIVE - Logarithms, Exponents, Limits, Course Project, Portfolio

(1) The invention of logarithms was one of the two or three most important developments in mathematics in the last 400 years. (the other two being calculus and non-Euclidean geometry

(2) Among their many uses, logarithms reduce the multiplication and division of very large numbers
to mathematical operations using only addition and subtraction of much smaller numbers.

(3) The use of logarithms to reduce multiplication and division to addition and subtraction is
how calculations are done using a slide rule.

(4) The logarithm of a number is the power to which you must raise some base in order to
generate the original number.

(5) For example, if I raise the base 10 to the third power, the result is 1000. The logarithm
of 1000 therefore (in base 10) is 3.

(6) In MathCad, the log_base_10 function is log and the natural log function is ln

(7) To calculate logarithms in any other base, you use the log function and you specify what base you want to use as part of the arguments. Therefore, the log_base_2 of 100 would be,..

(8) The "antilog" of any number is found by raising the base to the logarithm of the number.

(9) It is important to learn to manipulate this relationship between logarithms and exponents.

(10) The three most commonly used bases for logarithms are 10, 2, and the "natural log" the base of which is Euler's e.

(11) The base 10 is widely used because humans use the base 10 counting system.
The base 2 however is widely used in mathematics concerning information.

Can you imagine why?

(12) For all bases, taking the logarithm of the base itself equals the value one.

(13) Before programs like MathCad, conversion from one logarithmic base to another
was done using the following formula.

log_base_a (x) = log_base_b(x) / log_base_b(a)

(14) When solving some logarithms symbolically, MathCad will often convert the equation
automatically into base e using the formula above. For example,...

(15) As special case however is when you try to solve for the natural log of e^x.

(16) We can see why we obtain this solution if we pretend we are converting from one base,
say a, to another base b. According to the formula in line 13 above,..

log_base_a (x) = log_base_b(x) / log_base_b(a)

(18) In line 12 above however, we already stated that any logarithm of its own base = 1.
Therefore

(19) Euler's e (the base of the natural log) is defined as the limit of (1+1/x)^x as x
approaches infinity.

(20) MathCad has a built in value for Euler's e, but if we want to we can calculate our own
using the limit operator. In MathCad, this is called the "two sided limit operator".

(Note: the shortcut for this operator is ctrl + l )

(21) Just like the operators for integrals and derivatives, the limit operator requres that we
add in some information into the black boxes.

(22) The first piece of information is the formula which is typed into the black box to
the right of the word limit.

(23) The second piece of information is the variable that will be evaluated in this case, x.
Type x into the box on the left hand side of the arrow.

(24) The last piece of information is how far to take the equation. In limit operations it is
understood that the ideal would be to solve the limit for x = infinity.

(25) When limit processes were first developed however, great arguments raged over whether
one actually complete a limit process that went to infinity.

(26) Most mathematicians settled for solving the process for the largest value of x they could
stand to work out by hand. (I sometimes think this is the "limit" they keep talking about!)

(27) Even with the most powerful computer today, we still cannot solve any process for infinity. But MathCad can solve the limit for very large numbers, which will yield solutions of very high precision.

(28) So, back to calculating Euler's e. Even though MathCad really can't solve anything for infinity, MathCad does have a symbol for infinity on the calculus toolbar.

(29) I am still working on your project papers. I should be able to hand them back to you on Wednesday.

(30) However, you should not be sitting still waiting to get your project code back. Please
continue working on your projects.

(31) Remember, you will need to hand in your finished course project on Monday, April 24th.

(32) If you are still having trouble with your actual program come and see me this week!

(33) In addition to the course project, the due date for your Portfolio is coming up fast as well. The Portfolio will be due in class on Friday, April 28th.

(34) Your portfolio will consist of electronic copies of all your homework assignments as well
as an electronic copy of your course project.

(35) By "electronic copy" I mean the original MS Word document for assignments one through
eight and the MathCad documents you turned in for assignments nine and ten.

(36) You will turn in your portfolio on a 3.5" diskette. Your diskette must have a diskette label on it with your name, course number and the date.

(37) Your portfolio must be on a 3.5" diskette. Portfolios without a real diskette label will receive a grade of zero. (No tape. No postit notes. No packing labels. USE A DISKETTE LABEL.