Section 4.3 The Logarithmic Function

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Section 4.3The Logarithmic Function
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• Recall the definition of log
• In your groups see if you can draw the graph of
  log (without the aid of a calculator!!!)
• Here are some hints
• How are the graphs of y log x and x 10y
  related?
• If x 1, then what does the equation x 10y say
  that y has to be?
• If x lt 1. then according to the equation x 10y
  , what has to happen to y?
• If x gt 1. then according to the equation x 10y
  , what has to happen to y?
• If x is close to zero, then 10y has to be close
  to zero also. What does y have to do in order to
  get 10y close to zero?

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• Lets look at the graph of y log x together
• What is its vertical intercept?
• None
• What is its horizontal intercept?
• (1, 0)
• Does it have a horizontal asymptote?
• No, it continues increasing without bound
• Does it have a vertical asymptote?
• Yes, the line x 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 goes
  to positive or negative infinity
• The graph has a vertical asymptote at x a if
  the function goes to positive or negative
  infinity as x approaches a (from the right, left
  or both)

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• So what about the graph of y ln x?
• What is its vertical intercept?
• None
• What is its horizontal intercept?
• (1, 0)
• Does it have a horizontal asymptote?
• No, it continues increasing without bound
• Does it have a vertical asymptote?
• Yes, the line x 0
• In fact, y 10x (eln(10))x so the two graphs
  differ by a factor of ln(10)

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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?

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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 is measured in orders of 10
• These are known as orders of magnitude
• For example, the softest audible sound intensity
  is approximately 10-16 watts/cm2 (this is often
  denoted Io). The sound intensity of a
  refrigerator motor is 10-11 watts/cm2. Compare
  the two intensities.
• We do this by comparing the ratio of the two
  intensities
• The intensity of audible sounds has a very wide
  range so to scale down the numbers we use
  logarithm of the sound intensity, measured in
  decibels (dB)
• Noise levels in decibels

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Example

• A typical high school cafeteria has a sound
  intensity of 10-8 watts/cm2
• How many decibels is this?
• Recall that Io 10-16 watts/cm2 and dB
• Now suppose we have one sound measured at 65 dB
  and another measured at 50 dB. How many times
  more intense is the first sound than the second?
• Hint Create an equation for each intensity and
  use properties of logs

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