Logarithm

  • Definition: The logarithm of a number is the exponent to which another fixed value called the base must be raised to produce that number
  • The logarithm of x to the base a is denoted as loga(x)
  • Example: log2(8) = 3, as 2^3 = 8

Properties of Logarithms

  • Product Rule: loga(xy) = loga(x) + loga(y)
  • Quotient Rule: loga(x/y) = loga(x) - loga(y)
  • Power Rule: loga(x^n) = n*loga(x)
  • Change of Base Rule: loga(x) = logb(x) / logb(a)
  • Example: log2(4) = log10(4) / log10(2) = 2 / 0.3010 = 6.64

Common Logarithm

  • The common logarithm uses base 10, denoted as log(x) or log10(x)
  • Example: log(100) = 2, as 10^2 = 100

Natural Logarithm

  • The natural logarithm uses base e (Euler’s number), denoted as ln(x)
  • Example: ln(3) = 1.0986, as e^1.0986 = 3

Logarithmic Equations

  • Logarithmic equations involve logarithms and can be solved by exponentiating both sides
  • Example: Solve the equation log2(x) = 3. We have 2^3 = x, so x = 8

Logarithmic Inequalities

  • Logarithmic inequalities involve logarithms with inequality symbols, and their solutions may be intervals
  • Example: Solve the inequality log2(x) < 2. We have 2^2 > x, so x > 4

Solving Exponential Equations using Logarithms

  • Exponential equations can be solved by taking logarithms of both sides
  • Example: Solve the equation 3^x = 27. We have x = log3(27) = 3

Logarithmic Form of Complex Numbers

  • Complex numbers can be expressed in logarithmic form as well
  • Example: Write the complex number z = 4 + 3i in logarithmic form. We have z = log|z| + iarg(z) = log5 + i(53.13°)

Logarithmic Differentiation

  • Logarithmic differentiation is a technique to simplify the differentiation of complex functions
  • Example: Differentiate the function y = (x^3)(e^x). We have log(y) = 3log(x) + x, then differentiate both sides with respect to x

Logarithm - Example - Evaluation of logarithms

  • Example: Evaluate the logarithm log2(32). We can rewrite 32 as 2^5, so log2(32) = 5

Logarithm - Example - Solving logarithmic equations

  • Example: Solve the equation log3(x) = 2. Rewrite in exponential form: 3^2 = x, so x = 9
  • Example: Solve the equation log(x-4) = 1. Rewrite in exponential form: 10^1 = (x-4), so x = 14

Logarithm - Example - Solving exponential equations

  • Example: Solve the equation e^x = 5. Take the natural logarithm of both sides: ln(e^x) = ln(5), so x = ln(5)
  • Example: Solve the equation 2^(2x-3) = 8. Rewrite 8 as 2^3, so 2^(2x-3) = 2^3. Equating the exponents, we get 2x-3 = 3, solve for x

Logarithmic Inequalities - Example

  • Example: Solve the inequality log10(x-2) > 1. Rewrite in exponential form: 10^(log10(x-2)) > 10^1. Simplify, solve for x
  • Example: Solve the inequality log2(x+1) < log2(x). Rewrite in exponential form: 2^(log2(x+1)) < 2^(log2(x)). Simplify, solve for x

Logarithmic Functions - Domain and Range

  • The domain of logarithmic functions is the set of all positive real numbers
  • The range of logarithmic functions is the set of all real numbers
  • Example: The function y = log2(x) has a domain of (0, ∞) and a range of (-∞, ∞)

Logarithmic Functions - Graphs

  • The graph of a logarithmic function is a curve that approaches but never touches the x-axis
  • The graph gets steeper as the base of the logarithm increases
  • Example: The graph of y = log2(x) increases slowly at first, then becomes steeper as x increases

Laws of Logarithms

  • Logarithmic functions follow certain laws that can simplify calculations
  • Law 1: loga(1) = 0, as a^0 = 1
  • Law 2: loga(a) = 1, as a^1 = a
  • Law 3: loga(x*y) = loga(x) + loga(y)
  • Law 4: loga(x/y) = loga(x) - loga(y)
  • Law 5: loga(x^n) = n*loga(x)

Properties of Logarithms - Example

  • Example: Simplify log2(8) + log2(4) - log2(2). Using the laws of logarithms, we have log2(8*4/2) = log2(16) = 4

Change of Base Rule

  • The change of base rule allows us to rewrite logarithms in different bases
  • The rule states: loga(x) = logb(x) / logb(a)
  • Example: Rewrite log5(125) in base 10. Using the change of base rule, we have log10(125) / log10(5)

Logarithmic Functions - Transformations

  • Logarithmic functions can be transformed using the following operations:
    • Vertical Shift: y = loga(x) + c
    • Horizontal Shift: y = loga(x +/- d)
    • Vertical Stretch/Compression: y = a * loga(x)
  • Example: The graph of y = log2(x-1) + 2 is a vertical shift of 2 units up and a horizontal shift of 1 unit to the right

Logarithmic Functions - Applications

  • Logarithmic functions have many applications in various fields such as finance, biology, and computer science
  • Examples:
    • In finance, logarithms are used for calculating interest rates and investments
    • In biology, logarithmic growth is often modeled using logarithmic functions
    • In computer science, logarithms are used for analyzing algorithms and data structures

Logarithmic Functions - Inverse

  • Logarithmic functions and exponential functions are inverses of each other
  • If y = loga(x), then a^y = x
  • Example: If y = log2(8), then 2^y = 8

Logarithmic Functions - Solving Exponential Equations

  • Logarithmic functions can be used to solve exponential equations
  • Example: Solve the equation 2^x = 16 using logarithmic functions. Take the logarithm of both sides with base 2, so log2(2^x) = log2(16)

Common Logarithm - Example

  • Example: Find the value of log(1000). Since 1000 = 10^3, log(1000) = 3

Natural Logarithm - Example

  • Example: Find the value of ln(e^5). Since e^5 = 148.413, ln(e^5) = 5

Exponential Functions - Definition

  • Exponential functions have the form f(x) = a^x, where a is a positive constant
  • Example: The function f(x) = 2^x represents an exponential function

Exponential Growth and Decay - Formula

  • Exponential growth and decay can be modeled using the formula f(x) = a * e^(bx), where a and b are constants
  • Example: The function f(x) = 3 * e^(0.1x) represents exponential growth

Exponential Growth - Example

  • Example: A population of bacteria grows at a rate of 10% per hour. If the initial population is 1000, find the population after 5 hours using the formula P(t) = P(0) * e^(rt)

Exponential Decay - Example

  • Example: The value of a car depreciates at a rate of 5% per year. If the car is initially worth $20,000, find the value after 10 years using the formula V(t) = V(0) * e^(rt)

Logarithmic Functions - Practical Examples

  • Logarithmic functions are used in various practical applications, including:
    • pH scale: Logarithmic scale used to measure acidity or alkalinity
    • Sound intensity: Decibels (dB) use a logarithmic scale to measure sound intensity
    • Earthquake magnitude: Richter scale uses logarithms to measure earthquake strength

Summary

  • Logarithms are mathematical functions that represent the exponent to which a fixed value must be raised to produce another value
  • Logarithmic functions have many applications and are used to solve exponential equations
  • Exponential functions model growth and decay, while logarithmic functions model inverses
  • Understanding logarithms and exponential functions is important for various fields and practical applications