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Logarithmic Functions and Their Graphs

Logarithmic Functions and Their Graphs. is called the natural log function. is called the common log function. Consider. This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Two raised to what power is 16?. Example:.

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Logarithmic Functions and Their Graphs

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  1. Logarithmic Functions and Their Graphs

  2. is called the natural log function. is called the common log function. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Two raised to what power is 16? Example: The most commonly used bases for logs are 10: and e:

  3. Definition of Logarithmic Function b > 0; b  1 Logarithmic Form Exponential Form y = logb x x = by

  4. The log to the base “b” of “x” is the exponent to which “b” must be raised to obtain “x” y = log10 x  x = 10 y y = log e x  x = e y

  5. Change from Logarithmic To Exponential Form Log 2 8 = 3  8 = 23 Log 25 5 = ½  5 = 25 ½

  6. Change from Exponential To Form Logarithmic 49 = 7 2  log 7 49 = 2  log 5 (1/5) = -1 1/5 = 5 –1

  7. 3log x = x 3 Properties of Logarithmic Functions If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Log15 1 = 0 150 = 1 Log10 10 = 1 101 = 10 Log5 5x = x 5x = 5x

  8. The Decibel Scale The decibel level D of a sound of intensity I , measured in watts per square meter (W/ m2) is given by where I0 = 10–12 W/ m2 is the intensity of the least audible sound that an average healthy person can hear. Sound Intensity, W/ m2 Sound   1.0  10–12 Threshold of hearing 5.2  10–10 Whisper 3.2  10–6 Normal conversation 8.5  10–4 Heavy traffic 3.2  10–3 Jackhammer 1.0  100 Threshold of pain 8.3  102 Jet plane with afterburner

  9. The Richter Scale The magnitude M on the Richter scale of an earthquake that releases energy E , measured in joules, is given by where E0 = 104.40 joules is the energy released by a small reference earthquake. Magnitude on Richter scale Destructive power M < 4.5 Small 4.5 < M < 5.5 Moderate 5.5 < M < 6.5 Large 6.5 < M < 7.5 Major 7.5 < M Greatest

  10. Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged…

  11. y = 2x y = x –1 f f x = 2y x y x y = 2 x = 2 y 1 1 –3 –3 8 8 1 1 –2 –2 4 4 1 1 –1 –1 2 2 0 1 1 0 1 2 2 1 2 4 4 2 3 8 8 3 Ordered pairs reversed Logarithmic Function with Base 2 f y 10 f -1 5 or x y = log 2 x –5 5 10 –5 f DOMAINof = (– , ) = RANGE of f -1 f f -1 = (0, ) = DOMAIN of RANGE of

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