GM-Slides-Week-7-Logarithmic-Function-3


Learning Competencies

  • Represents real-life situations using logarithmic functions.

  • Distinguishes between logarithmic function, logarithmic equation, and logarithmic inequality.

  • Solves logarithmic equations and inequalities.

Basics of Logarithms

Exponential Form vs Logarithmic Form

  • Exponential form examples:

    • 2 = 8 implies log2(8) = 3

    • 5 = 25 implies log5(25) = 2

    • 8 = 1 implies log8(1) = 0

  • Important equations:

    • 3^4 = 81 implies log3(81) = 4

    • 6^2 = 36 implies log6(36) = 2

  • Logarithm Properties:

    • logb a is defined as c if and only if a = b^c

    • Logarithmic functions and exponential functions are inverses.

Key Definitions

  • Logarithm: Exponent to which the base must be raised to produce a given number.

  • Logarithmic Function: f(x) = logb(x) for b > 0 and b ≠ 1.

  • Logarithmic Equation: Involves logarithms (e.g., log2(16) = 4).

  • Logarithmic Inequality: Involves inequalities with logarithms (e.g., log2(x) > 3).

Properties & Laws of Logarithms

  1. Product Law: logb(MN) = logb(M) + logb(N)

    • Example: logb(3×5) = logb(3) + logb(5)

  2. Quotient Law: logb(M/N) = logb(M) - logb(N)

    • Example: logb(8/2) = logb(8) - logb(2)

  3. Power Law: logb(M^p) = p*logb(M)

    • Example: logb(2^3) = 3*logb(2)

Solving Logarithmic Equations

Techniques

  • Rewriting to exponential form: logb(a) = x → a = b^x

  • Using properties of logarithms to simplify

  • Applying the one-to-one property of logarithmic functions

  • Zero Factor Property: If ab = 0, then a = 0 or b = 0.

  • Considering the domain of logarithmic expressions

Examples

  • Solve log5(x + 3) = log5(22)

    • Leads to x + 3 = 22 → x = 19

  • Solve log3(9x) - log3(x-8) = 4

    • Leads to 9x/(x - 8) = 81 → Solve for x.

Solving Logarithmic Inequalities

  • If b > 0: logb(x) is an increasing function.

  • If 0 < b < 1: logb(x) is a decreasing function.

  • Solve inequalities similarly to equations.

Examples

  • log2(2x + 1) < 3 → leads to x > (7/2)

  • log3(2x-1) < 2 → leads to (1/2) < x < 5

Real-life Applications of Logarithms

Richter Scale

  • Magnitude of earthquakes is measured using a logarithmic scale: R = (1/3) * log(E/10^4.40)

    • Example: For an earthquake releasing 10^8 joules, R ≈ 2.4 (indicates release of 3981 times more energy than a reference quake).

Sound Intensity in Decibels

  • Decibel level: D = 10 log(I/I₀) where I₀ is the least audible sound.

    • Example: For a lawn mower at 10^-3 watts, D = 90 dB (1 billion times more intense than the threshold).

pH Scale

  • pH = -log[H+]

    • Example: pH of a solution with 0.01 moles of H+ is 2 (acidic).

Representations of Logarithmic Functions

Sketching Graphs

  • Graphs of logarithmic functions exhibit:

    • Vertical asymptote at x = 0.

    • Domain: (0, ∞)

    • Range: (-∞, ∞)

    • x-intercept at (1, 0).

Transformations of Logarithmic Functions

  • Horizontal shifts: f(x) = logb(x + c) shifts left (c > 0) or right (c < 0).

  • Vertical shifts: f(x) = logb(x) + d shifts up (d > 0) or down (d < 0).

  • Vertical stretches/compressions: f(x) = a logb(x) where a > 1 stretches and 0 < a < 1 compresses.

Practice and Application

  • Rewrite and convert log and exponential forms.

  • Evaluate logarithms and solve equations or inequalities as per the models above.