Logarithms and Their Properties

Definition and Properties

• A logarithm is the exponent to which a base must be raised to produce a given number. Mathematically, if b^y = x, then log_b(x) = y.
• The base 'b' is a positive real number not equal to 1.
• Common logarithms use base 10 (log₁₀(x)). Natural logarithms use base e (log_e(x)), which is also written as ln(x).
• Logarithms are the inverse functions of exponential functions. If f(x) = b^x, then f^-1(x) = log_b(x).

Logarithm Rules

• 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(x^y) = y * log_b(x)
• Change of base formula: log_b(x) = log_a(x) / log_a(b) (where 'a' is any base)
• log_b(1) = 0
• log_b(b) = 1

Common Logarithms (Base 10)

• The common logarithm, log(x), is the exponent to which 10 must be raised to equal x.
• Useful in scientific notation and problem solving in different scientific disciplines.

Natural Logarithms (Base e)

• The natural logarithm, ln(x), uses the irrational number e, approximately equal to 2.71828.
• ln(x) is the exponent to which e must be raised to equal x.
• Widely used in calculus and other areas of mathematics and science due to its relationship with exponential functions and derivatives.

Graphs

• Logarithmic graphs typically show a curve initially increasing rapidly and gradually flattening out as x grows larger.
• The graph always passes through the point (1,0) and approaches, but never touches, the y-axis.

Applications

• Widely used in various scientific fields including:
  • Physics: Analyzing decay rates of radioactive materials, measuring sound intensity, and studying earthquakes.
  • Biology: Modeling population growth and decay, and understanding chemical reactions.
  • Chemistry: Calculating pH levels, determining concentration and measuring acidity.
  • Engineering: Dealing with exponential growth, decay and other phenomena related to physics and engineering.
• The Richter scale, used to measure the magnitude of earthquakes, and decibels, used for measuring sound intensity, are also logarithmic scales.

Domains and Ranges

• The domain of a logarithmic function (log_b(x)) is all positive real numbers (x > 0).
• The range of a logarithmic function (log_b(x)) is all real numbers.

Calculating Logarithms

• Calculators and mathematical software are typically used to calculate logarithms.