MAT150 College Algebra

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• MAT150 College Algebra

Section 4.3 The Logarithm Function
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• Exponential Growth
• f(x)bx for b gt1
• Domain -8 lt x lt 8 Range 0 lt f(x) lt8
• Horizontal Asymptote y0, x-axis
• Vertical Intercept (0, 1)

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• Exponential Decay
• f(x)bx for 0 lt b lt1
• Domain -8 lt x lt 8 Range 0 lt f(x) lt8
• Horizontal Asymptote yo, x-axis
• Vertical Intercept (0, 1)

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• What is the equation of the green line?

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• Logarithmic Functions
• f(x)logax, agt1
• Domain 0 ltxlt8 Range -8 lt f(x) lt8
• Vertical Asymptote x 0, y-axis
• Horizontal Intercept (1,0)

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• Logarithmic Functions
• f(x)logax for 0 lt a lt 1
• Domain 0 ltxlt 8 Range -8 lt f(x) lt8
• Vertical Asymptote y-axis
• Horizontal Intercept (1,0)

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• Note
• The graph has a horizontal asymptote at y a if
  the function approaches the value of a as x
  approaches positive or negative infinity.
• f(x)?a as x?8 or as x?8
• The graph has a vertical asymptote at x a if
  the function increases (or decreases) without
  bound as x approaches a (from the right, left or
  both)
• f(x)?8 as x?a

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• Give the domain and range of each of the
  following, and sketch the graph
• a. F(x)ln(x)
• b. G(x)log(x 6)
• c. H(x)log(5 x)
• d. K(x)5(1.35)x
• e. P(x)2x7

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• Answer the following for f(x)logb(x).
• a. Give the domain of f(x)
• b. Give the range of f(x)
• c. For what values of x is f(x) gt 0?
• d. For what values of x is f(x) lt 0?
• e. What is the vertical intercept?
• f. What is the horizontal intercept?
• g. For what value of x is f(x)1?

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• Solving Logarithmic Equations Algebraically
• Step 1 Isolate the log
• Step 2 Convert to exponential form
• Step 3 Solve
• Step 4 Check
• NOTE! Do not round until the very end!!!!!!!
• Examples
• 4log5(x 3)8 2log(5x) 4 ln(3x1)7
•  

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Applications of Logs

• Richter Scale
• Used to measure the intensity of earthquakes
• Uses a base 10 logarithm to compare one
  earthquake to another.
• If City A has an earthquake of 2 on the Richter
  Scale, and City B has an earthquake of 4, it is
  not twice as powerful. In fact, it is ______
  times as powerful!
• What if City A2 and City B6? Is it 3 times as
  powerful?

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Applications of Logs

• Richter Scale
• Used to measure the intensity of earthquakes
• Uses a base 10 logarithm to compare one
  earthquake to another.
• Suppose an earthquake in Japan measures 4.8 on
  the Richter Scale, and an earthquake in India is
  twice the intensity in seismic units. What is
  the Richter Scale rating of the India earthquake?

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Applications of Logs

• Chemical Acidity (pH level)
• The acidity depends on the hydrogen ion
  concentration in the liquid (in moles per liter)
  written H.
• The greater the hydrogen ion concentration, the
  more acidic the solution
• The pH is defined as pH log H
• Pure water contains a hydrogen ion concentration
  of 1 x 10-7 moles. What is its pH?
• Find the hydrogen ion concentration of lemon
  juice, which has a pH of 2.3.

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Some facts about pH

• pH is used to measure the acidity of a liquid
• pH stands for power of hydrogen or potential
  hydrogen
• pH is a critical measurement. Life depends upon
  it. For instance, human blood is basic with a pH
  between 7.3 and 7.5. If the pH of blood drops
  below 7.3, acidosis occurs. If the blood pH rises
  above 7.5, alkalosis occurs. Death will occur if
  blood pH goes below 7.0 or above 7.8. Our human
  existence depends upon a balanced and buffered
  blood pH.

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• Sound Intensity (acoustic power)
• The human ear is very sensitive. The softest
  audible sound is approximately 10-12 watts/m2
  (this is often denoted Io). Sometimes called
  the threshold of hearing.
• Logarithms can help us cope with this huge range
  of values without losing track of the number of
  zeros after the decimal point.
• Noise levels in decibels

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Some Facts about Decibels

• The decibel scale is named after Alexander Graham
  Bell (thats why the B in the abbreviation dB is
  capitalized)
• A decibel is not a unit in the sense of meter or
  pound, it describes a relationship between two
  noise levels.

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Examples

• The sound of one person talking normally is about
  1.6x10-6 watts/cm2
• How many decibels is this?
• Suppose two people are talking. How many
  decibels is this?
• The noise level at a rock concert can measure
  120dB whereas the noise level at a library is
  about 31 decibels. How many times more intense
  is the noise at a rock concert than at a library?
• Hint Create an equation for each intensity and
  use properties of logs to find the difference.

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• Due next time
• Section 4.3 115 odd, 29, 31, 32
• Suggested review problems for Chapter 4
• (not assigned)
• pg. 185 1, 5, 9, 13ab, 15, 30, 31, 32, 33,
  41, 43, 45