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Logarithmic Functions

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1. 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

2. 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.

3. Examples Logarithmic Form Exponential Form 2x = 8 Two to what exponent is 8? 5x = 5 Five to what exponent is 5? 4x = Four to what exponent is ? 8

4. Examples Logarithmic Form 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” 100 We call “log” the “common log” 1 We call “ln” the “natural log”

5. Properties of Logarithms • 1 = 0 • a=1 • If x = y, then x= y • = x • = x

6. 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)

7. (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

8. Applications • The function 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)?

9. 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?

10. Applications • The loudness, L(x), measured in decibels, of a sound of intensity, x, measured in watts per square meter is defined as Determine the loudness, in decibels, of normal conversation intensity of x = 10-7 watt per square meter

11. Assignment • Page 411 #53, 55, 59-63 odd, 113, 115, 119