LOGARITHMS,MATRICES and COMPLEX NUMBERS

ByMaurice Fritz and Karen Greer

Let’s review

23 = 8 Base

2

Exponent

3

Power

What do you know about logarithms?

We call the exponent 3 the logarithm of 8 with base 2.  We write

3  =  log28.

WHEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 23.

23 = 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?

EXPONENTIAL TO LOGARITHM

The exponential equation 23 = 8 could be written in terms of a logarithmic equation as log2 8 = 3 .

23 = 8

log

2 8=3

which we say “log-base-2 of 8 equals 3”.

More about logarithm:

Given: log 100. Since there is no subscript

the base of 10 is automatically assumed.

Therefore log 10100 is 2.

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 23 .

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.

Convert 63 = 216 to the equivalent logarithmic expression.

Let’s do some conversion examples

Convert 33 = 27 to the equivalent logarithmic expression.

Solve: log5 25 = y.

5 to what exponent produces 25?

The required power is 2, because 52 = 25. Then 52 = 5 y = 25, so:log5(25) = 2; therefore, y = 2

Solve: log6 36 = y.

Solve: log2 32 = x.

Solve: log8 1 = x.

Solve: log7 7 = x.

The three laws of logarithms

1.    logb xy  =  logb x  +  logb 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 numeratorminus the logarithm of the denominator.”

3.    logb xn  =  n logb x

“The logarithm of a power of x is equal to the exponent of that power times the logarithm of x.”

y =  logb x  -  logb ylogb x

  logbxy  =  logbx  +  logby

  log7 9y  =  log7 9 +  log7 y

  log3 (2xy) =  log3 2 +  log3 x +  log3 y

y logb x   =  logb x  -  logb y

y log3 2  =  log3 2  -  log3 y

5 log7 3  =  log7 3 -  log7 5 

logb xn  =  n logbx

log2 x9  =  9 log2x

log7 25  =  5 log7 2

z log5 3 x9y     log5 3  +  log5 x9 +  log5 y -  log5 z

=  log5 3  +  9 log5 x +  log5 y -  log5 z

=

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.

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

4 0 7 3 7 6 6 -1 0 -2 5 2

A

The dimension of a Matrix

rowrowrowrow

This matrix A has four rows

columncolumncolumn

and three columns.

We say it is a 4 by 3 matrix.It is written as 4 X 3 matrix.

Rows x Columns

Operations with Matrices 4 -1 0

-2 5 3 A

8 2 6

-1 3 0B

Find A + B

4 8

Find A B

To add matrices just add the corresponding elements, the matrices being added must have the same dimensions.

Operations with Matrices 4 -1 0

-2 5 3 A

8 2 6 1 2 1 -1 3 2

B

Find AB

4 8

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.

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

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

Cramers Rule for 2 by 2 Systems

• This rule solves linear systems using • determinants. Given the system• a1 x + b1y = c1 and• a2x + b2y = c2

• We need 3 determinants as follow to solve• this. D = a1 b1 Dx = c1 b1 Dy = a1 c1

• a2 b2 c2 b2 a2 c2 • where c1 and c2 replace a1 and a2 in Dx• and c1 and c2 replace b1 and b2 in Dy.• The solution is x = Dx and y = Dy.• D D

• 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

• 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

•

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

Imaginary number

We don’t have a real number solution to this.

So we say,

i=And we call i an

i 3 = - ii 4 =1

i 2 =-1i 1 = i=i 0 =1

=-

Imaginary numbers

i 3 =- ii 4 =1i 2 =-1i 1 =ii 0 =1Imaginary numbers Continues

i 15 =i 4 i 4

i 4

i 3 =i 3 =- i

4 153 R3

i 38 =i 2 =- 1

4 389 R2

1 11

i 3

COMPLEX NUMBERS

What do you know about complex numbers?

A complex number is a number consisting of a real part and an imaginary part.

3 + 7i

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)i2

• Since i2 = -1, then we get 12i + 6(-1) =• -6 + 12i.

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.

• 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.

COMPLEX NUMBER EXERCISES

• 1 Find (a) i12 and (b) i23.• 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?