Topic: Logarithm - Introduction

  • Definition: Logarithm is the inverse operation of exponentiation.
  • Logarithm of a number x to the base b is denoted as log_b(x).
  • The base b must be a positive number and not equal to 1.
  • Logarithm helps us solve exponential equations.
  • Common logarithm: log_10(x) = log(x).
  • Natural logarithm: log_e(x) = ln(x).

Logarithmic Properties

  • log_b(xy) = log_b(x) + log_b(y)
  • log_b(x/y) = log_b(x) - log_b(y)
  • log_b(x^n) = n * log_b(x) (where n is a constant)
  • log_b(b^x) = x
  • log_b(1) = 0
  • log_b(b) = 1

Logarithmic Equations

  • Logarithmic equation: log_b(x) = y
  • Solve by converting to exponential form:
    • If log_b(x) = y, then b^y = x. Example: Solve for x - log_2(x) = 3 Convert to exponential form: 2^3 = x x = 8

Common Logarithm

  • Common logarithm uses base 10.
  • Denoted as log(x) or log_10(x).
  • Example: log(100) = 2 (since 10^2 = 100) Equation: log_b(x) = y
  • x is the value we want to find the logarithm of.
  • b is the base.
  • y is the exponent that we must raise the base to in order to get x.

Natural Logarithm

  • Natural logarithm uses base ’e'.
  • Denoted as ln(x).
  • Example: ln(e) = 1 Equation: ln(x) = y
  • e is a mathematical constant approximately equal to 2.71828.
  • x is the value we want to find the natural logarithm of.
  • y is the exponent that we must raise ’e’ to in order to get x.

Logarithmic Functions

  • Logarithmic functions are the inverse of exponential functions.
  • The general form of a logarithmic function is:
    • y = log_b(x) + c
    • c denotes the constant term. Example: Graph the function y = log_2(x) + 1
  • Plot the points for different x values and join them to get the graph.

Logarithm - Domain and Range

  • Domain of logarithmic function: (0, ∞)
  • The argument of the logarithm must be a positive real number. Example: Find the domain and range of the function y = log_3(x - 4)
  • Domain: x - 4 > 0
  • Domain: x > 4
  • Range: (-∞, ∞)

Logarithmic Functions - Transformations

  • Logarithmic functions can undergo transformations.
  • Vertical Shift: y = log_b(x) + c
  • Horizontal Shift: y = log_b(x - h)
  • Vertical Stretch/Compression: y = a * log_b(x) Example: Graph the function y = -log_2(x + 3) + 3
  • Apply vertical shift of 3 units up.
  • Apply horizontal shift of 3 units left.

Logarithmic Equations - Solve for x

  • Take logarithm of both sides of the equation.
  • Use logarithmic properties to simplify. Example: Solve for x - log_5(3x - 4) = 2
  • Take logarithm of both sides: log_5(3x - 4) = -2
  • Apply logarithmic property: 3x - 4 = 5^-2 Simplify and solve for x.

Logarithm - Applications

  • Logarithms have various applications in different fields:
  • Calculating population growth or decay rates.
  • Decibel scale for measuring sound.
  • pH scale for measuring acidity or alkalinity.
  • Earthquake magnitude scale.
  • Geometric growth and compound interest. Example: Solve the equation 2^x = 100 Take logarithm on both sides: log_2(2^x) = log_2(100) Simplify and solve for x.

Logarithm - Example - Problem on properties of logarithms

  • Evaluate the following expressions: a) log_2(8)

    b) log_3(1/27)

    c) log_5(√5)

  • Solution: a) log_2(8) = log_2(2^3) = 3

    b) log_3(1/27) = log_3(3^-3) = -3

    c) log_5(√5) = log_5(5^(1/2)) = 1/2

Logarithm - Example - Problem on solving logarithmic equations

  • Solve the following equation for x: log_2(x + 1) - log_2(x - 2) = 3

  • Solution: log_2(x + 1) - log_2(x - 2) = log_2((x + 1)/(x - 2)) = 3

    2^3 = (x + 1)/(x - 2)

    8(x - 2) = x + 1

    8x - 16 = x + 1

    7x = 17

    x ≈ 2.43

Logarithm - Example - Problem on logarithmic functions

  • Given the function y = log_4(x + 2), find the corresponding x-value for y = 2.

  • Solution: Setting y = 2, we have:

    2 = log_4(x + 2)

    4^2 = x + 2

    16 = x + 2

    x = 14

Logarithm - Example - Problem on domain and range

  • Find the domain and range of the function y = log_2(x - 3).

  • Solution: For the domain, we have:

    x - 3 > 0

    x > 3

    Domain: (3, ∞)

    For the range, since log_2(x - 3) is always positive, the range will be all positive real numbers.

    Range: (0, ∞)

Logarithm - Example - Problem on transformations

  • Given the function y = log_10(2x - 1), describe the transformations involved.

  • Solution: This function involves a horizontal shift to the right and a vertical stretch.

    • Horizontal shift: The function is shifted 1/2 units to the right.

    • Vertical stretch: The function is stretched vertically by a factor of 2.

Logarithm - Example - Problem on solving logarithmic equations

  • Solve the following equation for x: log_4(1 - 3x) = log_4(2 + x)

  • Solution: Setting the expressions inside log_4 equal to each other, we have:

    1 - 3x = 2 + x

    4x = -1

    x = -1/4

Logarithm - Example - Problem on logarithmic functions

  • Given the function y = -log_3(2x + 1), describe the transformations involved.

  • Solution: This function involves a horizontal shift to the left and a reflection about the x-axis.

    • Horizontal shift: The function is shifted 1/2 units to the left.

    • Reflection about the x-axis: The function is flipped upside down.

Logarithm - Example - Problem on logarithmic equations

  • Solve the following equation for x: 3log_5(x) + 2 = 1

  • Solution: Simplifying the equation, we have:

    3log_5(x) = -1

    log_5(x) = -1/3

    5^(-1/3) = x

    1/∛5 = x

Logarithm - Example - Problem on domain and range

  • Find the domain and range of the function y = log_3(4 - 2x).

  • Solution: For the domain, we have:

    4 - 2x > 0

    x < 2

    Domain: (-∞, 2)

    For the range, since log_3(4 - 2x) is always negative, the range will be all negative real numbers.

    Range: (-∞, 0)

Logarithm - Example - Problem on transformations

  • Given the function y = 3log_2(x) + 4, describe the transformations involved.

  • Solution: This function involves a vertical stretch and a vertical shift.

    • Vertical stretch: The function is stretched vertically by a factor of 3.

    • Vertical shift: The function is shifted 4 units up.

Logarithm - Example - Problem on properties of logarithms

  • Evaluate the following expressions: a) log_2(8)

    b) log_3(1/27)

    c) log_5(√5)

  • Solution: a) log_2(8) = log_2(2^3) = 3

    b) log_3(1/27) = log_3(3^-3) = -3

    c) log_5(√5) = log_5(5^(1/2)) = 1/2

Logarithm - Example - Problem on solving logarithmic equations

  • Solve the following equation for x: log_2(x + 1) - log_2(x - 2) = 3

  • Solution: log_2(x + 1) - log_2(x - 2) = log_2((x + 1)/(x - 2)) = 3

    2^3 = (x + 1)/(x - 2)

    8(x - 2) = x + 1

    8x - 16 = x + 1

    7x = 17

    x ≈ 2.43

Logarithm - Example - Problem on logarithmic functions

  • Given the function y = log_4(x + 2), find the corresponding x-value for y = 2.

  • Solution: Setting y = 2, we have:

    2 = log_4(x + 2)

    4^2 = x + 2

    16 = x + 2

    x = 14

Logarithm - Example - Problem on domain and range

  • Find the domain and range of the function y = log_2(x - 3).

  • Solution: For the domain, we have:

    x - 3 > 0

    x > 3

    Domain: (3, ∞)

    For the range, since log_2(x - 3) is always positive, the range will be all positive real numbers.

    Range: (0, ∞)

Logarithm - Example - Problem on transformations

  • Given the function y = log_10(2x - 1), describe the transformations involved.

  • Solution: This function involves a horizontal shift to the right and a vertical stretch.

    • Horizontal shift: The function is shifted 1/2 units to the right.

    • Vertical stretch: The function is stretched vertically by a factor of 2.

Logarithm - Example - Problem on solving logarithmic equations

  • Solve the following equation for x: log_4(1 - 3x) = log_4(2 + x)

  • Solution: Setting the expressions inside log_4 equal to each other, we have:

    1 - 3x = 2 + x

    4x = -1

    x = -1/4

Logarithm - Example - Problem on logarithmic functions

  • Given the function y = -log_3(2x + 1), describe the transformations involved.

  • Solution: This function involves a horizontal shift to the left and a reflection about the x-axis.

    • Horizontal shift: The function is shifted 1/2 units to the left.

    • Reflection about the x-axis: The function is flipped upside down.

Logarithm - Example - Problem on logarithmic equations

  • Solve the following equation for x: 3log_5(x) + 2 = 1

  • Solution: Simplifying the equation, we have:

    3log_5(x) = -1

    log_5(x) = -1/3

    5^(-1/3) = x

    1/∛5 = x

Logarithm - Example - Problem on domain and range

  • Find the domain and range of the function y = log_3(4 - 2x).

  • Solution: For the domain, we have:

    4 - 2x > 0

    x < 2

    Domain: (-∞, 2)

    For the range, since log_3(4 - 2x) is always negative, the range will be all negative real numbers.

    Range: (-∞, 0)

Logarithm - Example - Problem on transformations

  • Given the function y = 3log_2(x) + 4, describe the transformations involved.

  • Solution: This function involves a vertical stretch and a vertical shift.

    • Vertical stretch: The function is stretched vertically by a factor of 3.

    • Vertical shift: The function is shifted 4 units up.