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Logarithmic Functions • Categorized as Transcendental (Non-Algebraic) Functions • Inverse of an Exponential Function • Many real-life situations can be modeled using logarithmic functions, including: – Earthquake Intensity – Human Memory – Pace of life in Large Cities Definition of Logarithmic Function For x > 0 and a > 0, a≠ 1, = x if and only if x = ay This function, f(x) = x is called the logarithmic function with base a. Essentially, a logarithm is an exponent. Examples Logarithmic Form 2 8 5 5 4 1 16 Exponential Form 2x = 8 Two to what exponent is 8? 5x = 5 Five to what exponent is 5? 4x = 1 16 Four to what exponent is 1 ? 16 Examples Logarithmic Form 10 100 We call “log” the “common log” 1 We call “ln” the “natural log” Exponential Form 10x = 100 10 to what exponent is 100? We call “10” the “common base” ex = 1 e to what exponent is 1? We call “e” the “natural base” Properties of Logarithms • 1 = 0 • = x • a=1 • = x • If x = y, then x= y Logarithmic Function f(x) = • Domain (0, ∞) • Range (-∞, ∞) • No y-intercept since y-axis is vertical asymptote • x-intercept (1, 0) • Continuous • Neither even nor odd • Increasing (0, ∞) • No relative minimum or maximum • Has an inverse (exponential) X 1 2 4 8 y=f(x) = + (bx-c) • • • • • • • Transformations learned in Chapter 1 still apply Parent is exponential function with base a Vertical translation –”d” Horizontal translation –”bx-c=0” Reflection on x-axis – “sign of a” Reflection on y-axis-”sign of b” Vertical Stretch or Shrink – “numeric value of a” • EXAMPLES Applications • The function = 13.4 + 46.3 ln models the percentage of inbound email in the United States considered “spam”, f(x), where x is the number of years after 2000. According to this model, what percentage of email was considered spam in 2003 (round to nearest percent)? Applications • The percentage of adult height attained by a girl who is x years old can be modeled by f(x) = 62 +35log (x -4) where x represents the girl’s age (from 5-15) and f(x) represents the percentage of her adult height. According to the model, what percentage of her adult height has a girl attained at age 13, rounded to nearest tenth? Applications • The loudness, L(x), measured in decibels, of a sound of intensity, x, measured in watts per square meter is defined as = 10 −12 10 Determine the loudness, in decibels, of normal conversation intensity of x = 10-7 watt per square meter Assignment • Page 411 #53, 55, 59-63 odd, 113, 115, 119