Chapter 1: Introduction

• Reading week on June 25th, college closed

• Major quiz number 1 on June 6th

• Quiz available on My math lab on July 6th

• Quiz covers complex numbers and exponential/logarithmic functions

Chapter 13: Exponential Functions

• Exponential function defined as y = B^x

• B can be any number greater than 1 or between 0 and 1

• B cannot be equal to 1

• Examples: y = 2^x and y = 1/2^x

Graphs of Exponential Functions

• Fill table and graph y = 2^x and y = 1/2^x

• Values of y increase as x decreases for y = 2^x

• Values of y decrease as x increases for y = 1/2^x

Chapter 2: Graph Of Function

• Graphing the function

  • Consider the x and y axis

  • X axis is a straight line

  • Every two squares as 1 unit

  • Values on the y axis: -3, -2, -1, 0, 1, 2, 3

  • Largest value of y is 8

  • Plotting the points on the graph

• Properties of the exponential function

  • Domain: all real numbers

  • Range: values of y greater than 0

  • X axis is an asymptote

  • Function is increasing

• Graphing another function: 1 over 2 to the x

  • Calculating values for x

  • Plotting the points on the graph

Chapter 3: Replaced Log Base Base E We

• Exponential functions with base greater than 1 are increasing

• Exponential functions with base between 0 and 1 are decreasing

• Domain and range are the same

• X-axis is an asymptote for the function

Logarithmic Functions

• Exponential function: y = B^x

• Logarithmic form: x = log_B(y)

• Relation between logarithmic and exponential functions with the same base

Exercise: Converting Exponential Functions to Logarithmic Form

• Example 1: 5^2 = 25

  • Logarithmic form: log₅(25) = 2

• Example 2: 3^(-2) = 1/9

  • Logarithmic form: log₃(1/9) = -2

• Example 3: 81^(3/4) = 27

  • Logarithmic form: log₈₁(27) = 3/4

• Example 4: 2^(2+5x) = 3y

  • Logarithmic form: log₂(3y) = 2+5x

• Example 5: e^(x^2+2x+3) = 1+3y

  • Logarithmic form: ln(1+3y) = x^2+2x+3

Chapter 4: Base To Number

• Logarithm with no base given is logarithm based in 10

• Logarithmic form can be converted to exponential form

  • Example: log₁₀11 = 11² = 121

  • Example: log₈16 = 4/3 = 16

  • Example: log₇(1/49) = -2 = 1/49

  • Example: log₂(3/y-1) = 1+6x = (x²-6x+5)

  • Example: log₁₀(y+2) = x²-6x+5

• Unknown values can be determined by converting logarithmic form to exponential form

  • Example: log₅125 = 5^x = 5³ = 125

  • Unknown: x = 3

Chapter 5: Graph The Function

• Equation: n + 1 = 3

  • Convert to exponential form: 8^3 = n + 1

  • Isolate n: n = 8^3 - 1 = 511

• Equation: b^4 = 625

  • Raise both sides to the power of 1/4: b = 625^(1/4) = 5

• Equation: 5^(r + 1) = 5^2.3

  • Apply the rule of exponents: r + 1 = 2.3

  • Solve for r: r = 2.3 - 1 = 1.3

• Equation: b^(2/3) = 4

  • Raise both sides to the power of 3/2: b = 4^(3/2) = 8

Exercise 4: Graphing the Function

• Function: y = logarithm base 10 of x

• Values:

  • logarithm(1/4) = -0.6

  • logarithm(0.5) = -0.3

  • logarithm(1) = 0

  • logarithm(2) = 0.3

Chapter 6: Graph Of Function

• The values of the function are increasing

• The x-axis represents the x values and the y-axis represents the y values

• The graph intersects the x-axis at x=1

• The graph is close to the y-axis but never touches it

• The graph increases after x=1

• The properties of the logarithmic function are listed

• The domain of the logarithmic function is x>0

• The range of the logarithmic function is all real values of y

• The y-axis represents an asymptote to the graph of the logarithm

• The graph is negative when x is between 0 and 1, positive when x is greater than 1, and 0 when x=1

• The graph of the logarithmic function is the inverse of the exponential function

Chapter 7: Conclusion

• Next class will cover the properties of logarithmic functions

• These properties are important for solving problems

• Students are encouraged to review the properties before the next class

• More problems will be discussed in the next class

• Class will resume on Friday of week 7