Logarithm

• Definition of logarithm
• Logarithmic notation
• Exponential form
• Properties of logarithms
• Common logarithm
• Natural logarithm

Definition of Logarithm

• Logarithm is an operation that describes the number of times a given number (base) must be multiplied to produce another number (result).

Logarithmic Notation

• Logarithm of a number x to the base b is denoted as log_b(x).

Exponential Form

• The exponential form of b^y = x can be written as y = log_b(x).

Properties of Logarithms

• log_b(1) = 0
• log_b(b) = 1
• log_b(x * y) = log_b(x) + log_b(y)
• log_b(x / y) = log_b(x) - log_b(y)
• log_b(x^n) = n * log_b(x)

Common Logarithm

• Logarithm with base 10 is called common logarithm.
• It is denoted as log(x) or log_10(x).
• Example: log(100) = 2 because 10^2 = 100.

Natural Logarithm

• Logarithm with base e (Euler’s number) is called natural logarithm.
• It is denoted as ln(x) or log_e(x).
• Example: ln(e^2) = 2 because e^2 = 7.389.

Example - Simplification of Logarithms

• Given: log(4) + log(5)
• Using the property log_b(x * y) = log_b(x) + log_b(y)
• log(4 * 5) = log(20)
• Simplified form: log(20)

Equation Solving with Logarithms

• Logarithms can be used to solve equations involving exponentials.

Example - Equation Solving

• Given equation: 2^(3x - 1) = 8
• Taking both sides logarithm with base 2
• log_2(2^(3x - 1)) = log_2(8)
• Simplifying: 3x - 1 = 3
• Solving for x: 3x = 4
• Final answer: x = 4/3

Change of Base Formula

• Sometimes we need to calculate logarithms with bases that are not available in standard functions. Change of base formula helps us in these cases.

Change of Base Formula

• log_b(x) = log_c(x) / log_c(b)

Example - Change of Base Formula

• Given equation: log_3(2)
• We need to evaluate this using base 10.
• log_3(2) = log_10(2) / log_10(3)
• Using calculator: 0.6309 / 0.4771 = 1.3219

Graphs of Logarithmic Functions

• Logarithmic functions have a distinct shape in their graphs.

Graph of log(x)

• The graph of y = log(x) has the following properties:
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • X-intercept: (1, 0)
  • Asymptote: x-axis

Graph of ln(x)

• The graph of y = ln(x) has the following properties:
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • X-intercept: (1, 0)
  • Asymptote: x-axis

Logarithm - Example: Simplification of logarithms

• Given: log(4) + log(5)
• Using the property log_b(x * y) = log_b(x) + log_b(y)
• log(4 * 5) = log(20)
• Simplified form: log(20)

Equation Solving with Logarithms

• Logarithms can be used to solve equations involving exponentials.

Example: Equation Solving

• Given equation: 2^(3x - 1) = 8
• Taking both sides logarithm with base 2
• log_2(2^(3x - 1)) = log_2(8)
• Simplifying: 3x - 1 = 3
• Solving for x: 3x = 4
• Final answer: x = 4/3

Change of Base Formula

• Sometimes we need to calculate logarithms with bases that are not available in standard functions. Change of base formula helps us in these cases.

Change of Base Formula

• log_b(x) = log_c(x) / log_c(b)

Example: Change of Base Formula

• Given equation: log_3(2)
• We need to evaluate this using base 10.
• log_3(2) = log_10(2) / log_10(3)
• Using calculator: 0.6309 / 0.4771 = 1.3219

Graphs of Logarithmic Functions

• Logarithmic functions have a distinct shape in their graphs.

Graph of log(x)

• The graph of y = log(x) has the following properties:
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • X-intercept: (1, 0)
  • Asymptote: x-axis

Graph of ln(x)

• The graph of y = ln(x) has the following properties:
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • X-intercept: (1, 0)
  • Asymptote: x-axis

Logarithmic Equations

• Logarithmic equations involve the unknown variable in the argument of the logarithm.

Logarithmic Equation Example

• log(x - 1) = 2
• Solving for x:
  • Rewrite in exponential form: 10^2 = x - 1
  • Simplify: x = 100 + 1
  • Final answer: x = 101

Logarithmic Inequalities

• Logarithmic inequalities are inequalities that involve logarithmic expressions.

Logarithmic Inequality Example

• log(x + 1) > 2
• Solving for x:
  • Rewrite in exponential form: 10^2 < x + 1
  • Simplify: 100 < x + 1
  • Final answer: x > 99

Applications of Logarithms

• Logarithms are used in various fields and real-life situations.

Example: Compound Interest

• Compound interest formula: A = P(1 + r/n)^(nt)
• Logarithms can be used to solve for various unknowns in this formula.

Summary

• Logarithm is an operation used to describe the number of times a given number must be multiplied to produce another number.
• Logarithmic notation is denoted as log_b(x), where b is the base and x is the number.
• Logarithms have properties such as the product, quotient, and power properties.
• Common logarithm uses base 10, while the natural logarithm uses base e.
• Logarithmic equations and inequalities involve unknown variables within the logarithmic expressions.
• Logarithms have applications in finance, science, and various fields.

Logarithm - Example: Simplification of logarithms

• Given: log(4) + log(5)
• Using the property log_b(x * y) = log_b(x) + log_b(y)
• log(4 * 5) = log(20)
• Simplified form: log(20)

Equation Solving with Logarithms

• Logarithms can be used to solve equations involving exponentials.

Example: Equation Solving

• Given equation: 2^(3x - 1) = 8
• Taking both sides logarithm with base 2
• log_2(2^(3x - 1)) = log_2(8)
• Simplifying: 3x - 1 = 3
• Solving for x: 3x = 4
• Final answer: x = 4/3

Change of Base Formula

• Sometimes we need to calculate logarithms with bases that are not available in standard functions. Change of base formula helps us in these cases.

Change of Base Formula

• log_b(x) = log_c(x) / log_c(b)

Example: Change of Base Formula

• Given equation: log_3(2)
• We need to evaluate this using base 10.
• log_3(2) = log_10(2) / log_10(3)
• Using calculator: 0.6309 / 0.4771 = 1.3219

Graphs of Logarithmic Functions

• Logarithmic functions have a distinct shape in their graphs.

Graph of log(x)

• The graph of y = log(x) has the following properties:
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • X-intercept: (1, 0)
  • Asymptote: x-axis

Graph of ln(x)

• The graph of y = ln(x) has the following properties:
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • X-intercept: (1, 0)
  • Asymptote: x-axis

Logarithmic Equations

• Logarithmic equations involve the unknown variable in the argument of the logarithm.

Logarithmic Equation Example

• log(x - 1) = 2
• Solving for x:
  • Rewrite in exponential form: 10^2 = x - 1
  • Simplify: x = 100 + 1
  • Final answer: x = 101

Logarithmic Inequalities

• Logarithmic inequalities are inequalities that involve logarithmic expressions.

Logarithmic Inequality Example

• log(x + 1) > 2
• Solving for x:
  • Rewrite in exponential form: 10^2 < x + 1
  • Simplify: 100 < x + 1
  • Final answer: x > 99

Applications of Logarithms

• Logarithms are used in various fields and real-life situations.

Example: Compound Interest

• Compound interest formula: A = P(1 + r/n)^(nt)
• Logarithms can be used to solve for various unknowns in this formula.

Summary

• Logarithm is an operation used to describe the number of times a given number must be multiplied to produce another number.
• Logarithmic notation is denoted as log_b(x), where b is the base and x is the number.
• Logarithms have properties such as the product, quotient, and power properties.
• Common logarithm uses base 10, while the natural logarithm uses base e.
• Logarithmic equations and inequalities involve unknown variables within the logarithmic expressions.
• Logarithms have applications in finance, science, and various fields.