Topic: Logarithm

  • Definition of logarithm
  • Common logarithm
  • Natural logarithm
  • Properties of logarithms
  • Change of base formula

Definition of logarithm

  • The logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number.
  • Let’s consider a logarithm with base b, where b > 0 and b ≠ 1.
  • If b^x = y, then x is the logarithm of y to the base b, denoted as x = log_b(y).

Common logarithm

  • Common logarithm has a base of 10.
  • It is denoted as log(x) or log_10(x).
  • For example:
    • log(100) = 2, since 10^2 = 100.
    • log(1000) = 3, since 10^3 = 1000.

Natural logarithm

  • Natural logarithm has a base of e, where e is Euler’s number approximately equal to 2.71828182846.
  • It is denoted as ln(x).
  • For example:
    • ln(e) = 1, since e^1 = e.
    • ln(1) = 0, since e^0 = 1.

Properties of logarithms

  1. Product rule: log_b(xy) = log_b(x) + log_b(y)
  1. Quotient rule: log_b(x/y) = log_b(x) - log_b(y)
  1. Power rule: log_b(x^p) = p * log_b(x), where p is a real number
  1. Change of base formula: log_b(x) = log_c(x) / log_c(b), where c can be any positive value

Example: Product rule

  • Calculate log_2(4*8).
  • Using the product rule: log_2(4*8) = log_2(4) + log_2(8)
  • Simplifying the logarithms:
    • log_2(2^2) + log_2(2^3)
  • Applying the power rule:
    • 2 * log_2(2) + 3 * log_2(2)
  • Simplifying:
    • 2 + 3
  • Result: 5

Example: Quotient rule

  • Calculate log_5(125/5).
  • Using the quotient rule: log_5(125/5) = log_5(125) - log_5(5)
  • Simplifying the logarithms:
    • log_5(5^3) - log_5(5^1)
  • Applying the power rule:
    • 3 * log_5(5) - log_5(5)
  • Simplifying:
    • 3 - 1
  • Result: 2

Example: Power rule

  • Calculate log_3(27^2).
  • Using the power rule: log_3(27^2) = 2 * log_3(27)
  • Simplifying the logarithm:
    • 2 * log_3(3^3)
  • Applying the power rule:
    • 2 * 3
  • Result: 6

Change of base formula

  • The change of base formula allows us to convert logarithms from one base to another.
  • For example, to find log_2(8) using log_10, we can apply the formula:
    • log_2(8) = log_10(8) / log_10(2)
  • Using log properties, we can simplify the equation further.

Recap

  • Logarithm is the exponent to which a base must be raised to produce a given number.
  • Common logarithm has base 10, denoted as log(x).
  • Natural logarithm has base e, denoted as ln(x).
  • Logarithms have various properties like product rule, quotient rule, and power rule.
  • Change of base formula allows conversion of logarithms from one base to another.

Applications of Logarithms

  • Logarithms have many practical applications in various fields such as:
    • Engineering: Logarithms are used in signal processing, electrical circuit analysis, and control systems.
    • Finance: Logarithms are used in compound interest calculations and investment strategies.
    • Biology: Logarithms are used in pH calculations and population studies.
    • Computer Science: Logarithms are used in algorithms, data structures, and cryptography.

Solving Exponential Equations

  • Logarithms can be used to solve exponential equations.
  • To solve an equation of the form b^x = y, we can take the logarithm of both sides: x = log_b(y).
  • This allows us to solve for the unknown variable x.

Example:

  • Solve for x: 2^x = 16
  • Taking the logarithm of both sides: x = log_2(16)
  • Simplifying using the change of base formula: x = log(16) / log(2)
  • Evaluating the logarithms: x = 4 / 0.3010
  • Result: x ≈ 13.29

Laws of Logarithms

  • Logarithms follow certain laws or rules that help in simplifying complex equations involving logarithms.
  • Let’s discuss some of these laws:

Law of Logarithm of Product:

  • log_b(xy) = log_b(x) + log_b(y)

Law of Logarithm of Quotient:

  • log_b(x/y) = log_b(x) - log_b(y)

Law of Logarithm of Power:

  • log_b(x^p) = p * log_b(x), where p is a real number

Change of Base Formula

  • The change of base formula allows us to convert logarithms from one base to another.
  • The formula is as follows: log_b(x) = log_c(x) / log_c(b), where c is any positive value.
  • This formula is useful when working with logarithms of different bases.

Example:

  • Find log_2(8) using log_10.
  • Applying the change of base formula: log_2(8) = log_10(8) / log_10(2)
  • Evaluating the logarithms: log_2(8) ≈ 3 / 0.3010
  • Result: log_2(8) ≈ 9.97

Solving Logarithmic Equations

  • Logarithmic equations can also be solved using logarithmic properties and algebraic techniques.
  • We can convert logarithmic equations into exponential form to solve them.
  • Let’s look at an example:

Example:

  • Solve for x: log_2(x) + log_2(x + 2) = 3
  • Converting into exponential form: 2^3 = x(x + 2)
  • Simplifying: 8 = x^2 + 2x
  • Rearranging: x^2 + 2x - 8 = 0
  • Factoring: (x + 4)(x - 2) = 0
  • Solving for x: x = -4 or x = 2
  • Check for extraneous solutions: log_2(-4) and log_2(-2) are not defined, so x = 2 is the solution.

Logarithmic Properties - Continued

  • Let’s discuss some more logarithmic properties:

Law of Logarithm of Base:

  • log_b(b) = 1

Law of Logarithm of One:

  • log_b(1) = 0

Law of Logarithm of Identity:

  • log_b(b^x) = x

Law of Logarithm of Inverse:

  • log_b(b^(-x)) = -x

Law of Logarithm of Same Base:

  • log_b(b) = 1

Logarithms in Calculus

  • Logarithmic functions also play a significant role in calculus.
  • Logarithmic differentiation is used to simplify the differentiation of complex functions.
  • Logarithmic integration is used to solve integrals involving logarithmic functions.

Example: Logarithmic Differentiation

  • Calculate the derivative of y = (x^3)(ln(x))
  • Using logarithmic differentiation: ln(y) = ln(x^3) + ln(ln(x))
  • Taking the derivative of both sides: (1/y)(dy/dx) = (3/x) + (1/x) * (1/ln(x))
  • Simplifying: dy/dx = y * [(3/x) + (1/x) * (1/ln(x))]
  • Substituting: dy/dx = (x^3)(ln(x)) * [(3/x) + (1/x) * (1/ln(x))]

Common Logarithmic Identities

  • Here are some common logarithmic identities:

Identity 1:

  • log_b(x) = 1 / log_x(b)

Identity 2:

  • log_b(1 / x) = -log_b(x)

Identity 3:

  • log_b(x * y) = log_b(x) + log_b(y)

Identity 4:

  • log_b(x / y) = log_b(x) - log_b(y)

Identity 5:

  • log_b(x^r) = r * log_b(x), where r is a real number

Example: Using Logarithmic Identities

  • Evaluate the expression: log_2(8) / log_2(4) Using logarithmic identities:
  • log_2(8) / log_2(4) = (1 / log_8(2)) / (1 / log_4(2))
  • Simplifying: log_4(2) / log_8(2)
  • Expressing the bases in terms of a common base: log_2(2) / log_2(8^(1/3))
  • Simplifying further: 1 / (1/3) = 3 Result: log_2(8) / log_2(4) = 3

Recap

  • Logarithms have various applications in different fields.
  • Logarithmic equations can be solved using logarithmic properties and algebraic techniques.
  • Logarithmic differentiation and integration are important in calculus.
  • Logarithmic identities help simplify complex logarithmic expressions.
  • Understanding logarithms and their properties is crucial for solving problems and working with logarithmic functions.

Logarithm - Example

  • Problem:
    • Simplify the expression: log_3(27) + log_2(8) - log_4(16)
  • Solution:
    • Using the power rule: log_3(3^3) + log_2(2^3) - log_4(4^2)
    • Simplifying the logarithms: 3 + 3 - 2
  • Result: log_3(27) + log_2(8) - log_4(16) = 4

Logarithm - Example

  • Problem:
    • Solve for x: log_5(x^2) = 4
  • Solution:
    • Converting into exponential form: 5^4 = x^2
    • Simplifying: x^2 = 625
  • Taking the square root of both sides: x = ±25
  • Result: log_5(x^2) = 4 if x = ±25

Logarithm - Example

  • Problem:
    • Simplify the expression: log(e^(5lnx))
  • Solution:
    • Using the logarithmic identity: log_b(b^x) = x
    • Simplifying the logarithm: 5lnx
  • Result: log(e^(5lnx)) = 5lnx

Logarithm - Example

  • Problem:
    • Solve for x: log_2(3x + 2) + log_2(2x - 1) = log_2(20)
  • Solution:
    • Applying the product rule: log_2[(3x + 2)(2x - 1)] = log_2(20)
    • Simplifying the logarithms: (3x + 2)(2x - 1) = 20
    • Expanding: 6x^2 - x - 2 = 20
    • Rearranging: 6x^2 - x - 22 = 0
    • Factoring or using the quadratic formula to solve for x
  • Result: x ≈ -2.30 or x ≈ 1.90

Logarithm - Example

  • Problem:
    • Find the value of x: 2^(3x + 1) = 4^(2x - 3)
  • Solution:
    • Using the power rule: (2^3)^x * 2 = (4^2)^x * 4^(-3)
    • Simplifying the powers: 8^x * 2 = 16^x * 1/64
    • Simplifying further: (8/16)^x = 2^-6
    • Evaluating the fractions: 1/2^x = 1/64
    • Comparing the exponents: x = 6
  • Result: 2^(3x + 1) = 4^(2x - 3) if x = 6

Logarithm - Example

  • Problem:
    • Solve for x: log_3(3x + 2) + log_3(x - 1) = 2
  • Solution:
    • Applying the product rule: log_3[(3x + 2)(x - 1)] = 2
    • Simplifying the logarithms: (3x + 2)(x - 1) = 3^2
    • Expanding: 3x^2 - x - 2 = 9
    • Rearranging: 3x^2 - x - 11 = 0
    • Factoring or using the quadratic formula to solve for x
  • Result: x ≈ -1.48 or x ≈ 2.15

Logarithm - Example

  • Problem:
    • Simplify the expression: log_2(16) * log_5(125) - log_7(49)
  • Solution:
    • Using the power rule: log_2(2^4) * log_5(5^3) - log_7(7^2)
    • Simplifying the logarithms: 4 * 3 - 2
  • Result: log_2(16) * log_5(125) - log_7(49) = 10

Logarithm - Example

  • Problem:
    • Solve for x: log_5(4x + 3) = log_5(2x - 1) + 1
  • Solution:
    • Applying the quotient rule and logarithmic identity: log_5[(4x + 3) / (2x - 1)] = 1
    • Simplifying the quotient: (4x + 3) / (2x - 1) = 5^1
    • Simplifying further: (4x + 3) / (2x - 1) = 5
    • Cross-multiplying and solving for x
  • Result: x ≈ 2.20

Logarithm - Example

  • Problem:
    • Solve for x: log_3(7x + 1) = 2 - log_3(x - 2)
  • Solution:
    • Rearranging the equation: log_3(7x + 1) + log_3(x - 2) = 2
    • Applying the sum-to-product rule: log_3[(7x + 1)(x - 2)] = 2
    • Simplifying the logarithm: (7x + 1)(x - 2) = 3^2
    • Expanding: 7x^2 - 13x - 2 = 9
    • Rearranging: 7x^2 - 13x - 11 = 0
    • Factoring or using the quadratic formula to solve for x
  • Result: x ≈ -0.33 or x ≈ 2.18

Recap

  • Logarithm is a powerful mathematical concept used in various fields.
  • Logarithms can be simplified, solved, and used in equations and functions.
  • Understand logarithmic properties and identities to manipulate and solve logarithmic expressions.
  • Practice solving logarithmic equations and simplifying logarithmic expressions.
  • Logarithms are essential for understanding exponential growth, rate of change, and complex calculation.