1. LOGARITHMS,MATRICES and COMPLEX NUMBERS By Maurice Fritz and Karen Greer

2. Let’s review 2 3 = 8 Base 2 Exponent 3 Power

3. What do you know about logarithms?

4. We call the exponent 3 the logarithm of 8 with base 2. We write 3 = log 2 8. WHEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 2 3. 2 3 = 8 ? On the other hand, IF WE ARE GIVEN, the base 2 and its power 8 – That unknown exponent is called a logarithm. 2 ? = 8 -- then what is the exponent that will produce 8?

5. EXPONENTIAL TO LOGARITHM The exponential equation 2 3 = 8 could be written in terms of a logarithmic equation as log 2 8 = 3. 2 3 = 8 log 2 8 = 3 which we say “log-base-2 of 8 equals 3”.

6. More about logarithm: Given: log 100. Since there is no subscript the base of 10 is automatically assumed. Therefore log 10 100 is 2.

7. Multiplication is a shortcut for addition. For example: 3x5 means 5 + 5 + 5. Exponents are a shortcut for multiplication with the same number. For example: 2x2x2 means 2 3. Logarithm is a shortcut for exponents. Just as subtraction is the opposite of addition and division is the opposite of multiplication, Logarithms are the "opposite" of exponentials.

8. Convert 6 3 = 216 to the equivalent logarithmic expression. Let’s do some conversion examples Convert 3 3 = 27 to the equivalent logarithmic expression.

9. Solve: log 5 25 = y. 5 to what exponent produces 25? The required power is 2, because 5 2 = 25. Then 5 2 = 5 y = 25, so: log 5 (25) = 2; therefore, y = 2 Solve: log 6 36 = y. Solve: log 2 32 = x. Solve: log 8 1 = x. Solve: log 7 7 = x.

10. The three laws of logarithms 1. log b xy = log b x + log b y “The logarithm of a product is equal to the sum of the logarithms of each factor.” 2. “The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.” 3. log b x n = n log b x “The logarithm of a power of x is equal to the exponent of that power times the logarithm of x.” y = log b x - log b y log b x

11. log b xy = log b x + log b y log 7 9y = log 7 9 + log 7 y log 3 (2xy) = log 3 2 + log 3 x + log 3 y

12. y log b x = log b x - log b y y log 3 2 = log 3 2 - log 3 y 5 log 7 3 = log 7 3 - log 7 5

13. log b x n = n log b x log 2 x 9 = 9 log 2 x log 7 2 5 = 5 log 7 2

14. z log 5 3 x 9 y log 5 3 + log 5 x 9 + log 5 y - log 5 z = log 5 3 + 9 log 5 x + log 5 y - log 5 z =

15. What are applications of logarithms? 1. Logarithms are used in applications which are related to hearing because we hear with our ears in a logarithmic relation. This also has to do with equipment with decibel ratings because of how we hear. 2. Logarithms have been used for slide rules when multiplication and division are performed. When we perform slide rule addition, it is like the product rule for logarithms and results in multiplication. When we perform slide rule subtraction, it is like the quotient rule for logarithms and results in division.

16. In mathematics, a matrix (plural matrices) is a rectangular two-dimensional group of numbers, An item in a matrix is called an entry or an element. 4 0 7 3 7 6 6 -1 0 -2 5 2 A such as

17. 4 0 7 3 7 6 6 -1 0 -2 5 2 A The dimension of a Matrix row This matrix A has four rows column and three columns. We say it is a 4 by 3 matrix. It is written as 4 X 3 matrix. Rows x Columns

18. Operations with Matrices 4 -1 0 -2 5 3 A 8 2 6 -1 3 0 B Find A + B 48 Find A B To add matrices just add the corresponding elements, the matrices being added must have the same dimensions.

19. Operations with Matrices 4 -1 0 -2 5 3 A 8 2 6 1 2 1 -1 3 2 B Find AB 48 The product of two matrices is found by multiplying the elements of the rows of the matrix on the left by the corresponding elements of the columns of the matrix on the right. The resulting matrix must have the same column of the first one and row of the second one for the first row and the same row of the first one and column of the second for the second row.

20. A determinant is also an ordered array of numbers within rectangular brackets and has a lot of similarities to a matrix. The most useful application of determinants is in solving linear Algebraic equations. Solution of a 2 by 2 determinant Determinants

21. For a 2×2 matrix, its determinant is found by subtracting the products of its diagonals. Find a b = c d ad -bc Example: Find the following determinant. Find 2 6 = 7 5 (2)(5) - (6)(7) = 10 – 42 = -32

22. Cramers Rule for 2 by 2 Systems This rule solves linear systems using determinants. Given the system a 1 x + b 1 y = c 1 and a 2 x + b 2 y = c 2 We need 3 determinants as follow to solve this. D = a 1 b 1 Dx = c 1 b 1 Dy = a 1 c 1 a 2 b 2 c 2 b 2 a 2 c 2 where c 1 and c 2 replace a 1 and a 2 in Dx and c 1 and c 2 replace b 1 and b 2 in Dy. The solution is x = Dx and y = Dy. D D

23. Solve the following system using Cramer’s Rule and the necessary determinants. 2x - 3y = 2 5x + y = 22 Determinant D = 2 -3 = 2(1) –(-3)(5) = 17 5 1 Determinant Dx = 2 -3 = 2(1) –(-3)22 =68 22 1 Determinant Dy = 2 2 = 2(22) -5(2) = 34 5 22 X = Dx/D = 68/17 = 4 y = Dy/D = 34/17 = 2 Check 2(4) -3(2) = 8-6 = 2 5(4) + (1)2 = 20 +2 =22

24. Now lets practice what we have learned on some exercises with matrices and Determinants. 1 _______(A) True, (B) False. You can add a 2 by 3 matrix and a 3 by 4 matrix. 2 Given A = 5 -4 7 and B = 7 8 -2 6 2 -3 4 9 1 Find A + B 3 Given A = 3 4 2 and B = -6 4 5 0 4 2 3 3 -2 Find the product AB

25. 4 Find (a) D, (b) Dx,( c) Dy,(d) x and (e)y. for these 2 equations 3x + 2y = 9 7x - y = 4

26. Imaginary number We don’t have a real number solution to this. So we say, i = And we call i an

27. i 3 = - i i 4 =1 i 2 =-1 i 1 = i = i 0 =1 = - Imaginary numbers

28. i 3 = - i i 4 =1 i 2 =-1 i 1 =i i 0 =1 Imaginary numbers Continues i 15 =i4i4 i4i4 i4i4 i3i3 = i 3 = - i 415 3 R3 i 38 = i 2 = - 1 438 9 R2 1 11 i3i3

29. COMPLEX NUMBERS What do you know about complex numbers? A complex number is a number consisting of a real part and an imaginary part.number realimaginary 3 + 7 i

30. ADDING COMPLEX NUMBERS Example 1 Add A = (2 + 3i) + B = (5 + 4i). Add the real parts and the imaginary parts to get (2 + 5) + (3 + 4) i = 7 + 7i Example 2 Add A = (4 + 2i) + B = (3 -5i). (4 + 3) + (2 -5)i = 7 – 3i MULTIPLYING COMPLEX NUMBERS Multiply 3i(4 + 2i) = 3(4)i + (3)(2)i 2 Since i 2 = -1, then we get 12i + 6(-1) = -6 + 12i.

31. A PRACTICAL APPLICATION OF COMPLEX NUMBERS Complex numbers are used in electrical and electronic circuits. The real part is the resistive component. The imaginary parts are inductors which are +i and capacitors which are -i. Example: A 10 ohm resistor, a 15 ohm inductor and a 20 ohm capacitor are placed in a series circuit.

32. In a series circuit the complex number values need to be added. Find the resulting complex number value. 10 + 15i -20i = 10 -5i. The result is 10 ohms resistive + 5 ohms capacitive.

33. COMPLEX NUMBER EXERCISES 1 Find (a) i 12 and (b) i 23. 2 Add = (3 + 4i) + (2 -7i). 3 Multiply = 6i(4 -5i). 4 A 20 ohm resistor, a 30 ohm inductor and a 10 ohm capacitor are placed in a series circuit. Find the resulting complex number value. Is it resistive and inductive or resistive and capcitive?