Logarithm - Introduction to Logarithms

• Definition of logarithm: a logarithm is the exponent to which a base number must be raised to obtain a given number.
• Logarithm notation: Log_b(x)
• The base, b, can be any positive number except 1.
• The number x is called the argument of the logarithm. Example: Log₂(8) = y means 2^y = 8
• Properties of logarithms:
  • Logarithm of a product: Log_b(xy) = Log_b(x) + Log_b(y)
  • Logarithm of a quotient: Log_b(x/y) = Log_b(x) - Log_b(y)
  • Logarithm of a power: Log_b(xⁿ) = n * Log_b(x) Example: Log₂(8) = 3 and Log₂(4) = 2. Using the properties of logarithms, Log₂(64) = Log₂(8²) = 2 * Log₂(8) = 2 * 3 = 6
• Change of base formula: Log_b(x) = Log_a(x) / Log_a(b) Example: Calculate Log₃(7) using the change of base formula with base 10. Log₃(7) = Log₁₀(7) / Log₁₀(3)
• Natural logarithm: Log_e(x) = ln(x), where e is Euler’s number approximately equal to 2.71828 Example: ln(e) = 1 "

11. Rational Exponents

• Definition of rational exponents: rational exponents are exponents that are expressed as fractions.
• Relation between rational exponents and radicals: x^m/n = n√(x^m)
• Example: 3^2/3 = ∛(3²) = ∛9

12. Laws of Exponents with Rational Exponents

• Product of powers rule: (x^m/n)(x^p/n) = x^(m+p)/n
• Quotient of powers rule: (x^m/n)/(x^p/n) = x^(m-p)/n
• Power of a power rule: (x^m/n)^p/n = x^(m*p)/n Example: (4^1/2)(4^3/2) = 4^(1+3)/2 = 4^4/2 = 4² = 16

13. Solving Equations with Rational Exponents

• To solve an equation with rational exponents, rewrite the equation without fractional exponents by raising both sides of the equation to the same power.
• Example: Solve 2^x/3 = 8
  • Rewrite as (2^x/3)³ = 8³
  • Simplify to 2^x = 512
  • Solution is x = 9

14. Exponential Growth and Decay

• Exponential growth: a quantity increasing exponentially over time.
• Exponential decay: a quantity decreasing exponentially over time.
• General exponential growth or decay function: f(t) = P(1 + r/n)^nt or f(t) = P(1 - r/n)^nt
  • P: initial quantity
  • r: growth or decay rate
  • n: number of times interest is compounded per time period
  • t: time in years

15. Compound Interest

• Compound interest formula: A = P(1 + r/n)^nt
  • A: total amount after time t
  • P: principal amount
  • r: annual interest rate
  • n: number of times interest is compounded per year
  • t: time in years Example: Calculate the compound interest for a principal amount of $5000 with an annual interest rate of 4% compounded monthly over 3 years. A = 5000(1 + 0.04/12)^(12*3) = $5657.35

16. Logarithmic Functions

• Definition of logarithmic function: a function that is the inverse of an exponential function.
• Logarithmic function notation: y = log_b(x)
• The graph of a logarithmic function is the reflection of the graph of the exponential function over the line y = x.

17. Properties of Logarithmic Functions

• Logarithmic function of a product: log_b(xy) = log_b(x) + log_b(y)
• Logarithmic function of a quotient: log_b(x/y) = log_b(x) - log_b(y)
• Logarithmic function of a power: log_b(xⁿ) = n * log_b(x)

18. Solving Exponential Equations using Logarithms

• To solve an exponential equation, take the logarithm of both sides of the equation and use the properties of logarithms to simplify.
• Example: Solve 2^x = 16
  • Take the logarithm of both sides: log₂(2^x) = log₂(16)
  • Simplify: x = log₂(16) = 4

19. Common Logarithm

• Common logarithm: logarithm with base 10 (log₁₀(x) = log(x))
• Common logarithm notation: log(x) Example: log(100) = 2

20. Natural Logarithm

• Natural logarithm: logarithm with base e, denoted as ln(x)
• ln(x) = log_e(x) Example: ln(e²) = 2

21. Properties of Logarithmic Functions (continued)

• Change of base property: log_b(x) = log_a(x) / log_a(b)
• Common logarithm property: log_b(b) = 1
• Natural logarithm property: ln(e) = 1 Example: log₂(2) = 1 and ln(e) = 1

22. Solving Logarithmic Equations

• To solve a logarithmic equation, write the equation in exponential form and then solve for the variable.
• Example: Solve log₃(x) = 2
  • Rewrite as 3² = x
  • Simplify: x = 9

23. Applications of Logarithms - pH Scale

• The pH scale is a logarithmic scale used to measure the acidity or alkalinity of a solution.
• pH = -log₁₀([H+])
  • [H+]: concentration of hydrogen ions in moles per liter Example: If the hydrogen ion concentration of a solution is 0.001 M, calculate the pH. pH = -log₁₀(0.001) = 3

24. Applications of Logarithms - Sound Intensity

• Sound intensity is measured in decibels (dB), which is a logarithmic scale.
• The formula to calculate sound intensity in decibels is: L = 10 log₁₀(I/I₀)
  • L: sound level
  • I: sound intensity
  • I₀: reference sound intensity (usually set at the threshold of hearing) Example: If the sound intensity is 0.001 W/m², calculate the sound level in decibels using a reference intensity of 10^-12 W/m². L = 10 log₁₀(0.001 / 10^-12) = 120 dB

25. Applications of Logarithms - Richter Scale

• The Richter scale is used to measure the magnitude of earthquakes.
• The formula to calculate the magnitude on the Richter scale is: M = log₁₀(A/T) + b
  • M: Richter magnitude
  • A: maximum amplitude of seismic waves
  • T: period of seismic vibrations
  • b: constant value Example: If the maximum amplitude of seismic waves is 1 mm and the period of seismic vibrations is 10 seconds, calculate the magnitude on the Richter scale. M = log₁₀(0.001 / 10) + b

26. Logarithmic Identities

• Product rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient rule: log_b(x/y) = log_b(x) - log_b(y)
• Power rule: log_b(xⁿ) = n * log_b(x)
• Change of base rule: log_b(x) = log_a(x) / log_a(b) Example: log₂(8) = log₂(4) + log₂(2) = 2 + 1 = 3

27. Common Logarithmic Function Graph

• The graph of a common logarithmic function, y = log(x), has an asymptote at x = 0.
• The range of the function is all real numbers.
• The graph is increasing as x increases.

28. Natural Logarithmic Function Graph

• The graph of a natural logarithmic function, y = ln(x), is similar to the common logarithmic function.
• The graph has an asymptote at x = 0.
• The range of the function is all real numbers.
• The graph is increasing as x increases.

29. Exponential Functions and Logarithmic Functions

• Exponential functions and logarithmic functions are inversely related.
• The exponential function, y = b^x, and the logarithmic function, y = log_b(x), are reflections of each other over the line y = x.

30. Review and Practice Problems

• Are you ready for the quiz?
• Practice problems:
  1. Solve log₃(x) = 4
  2. Calculate the pH given a hydrogen ion concentration of 0.0001 M.
  3. Convert a sound intensity of 10^-9 W/m² to decibels using a reference intensity of 10^-12 W/m².
  4. Calculate the magnitude on the Richter scale given a maximum amplitude of seismic waves of 2 mm and a period of 8 seconds.
  5. Graph the function y = log₂(x) and find its domain and range.