Logarithms Continued

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Logarithms (Continued)

• GE 111 - Lecture 2

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Exponentials and Logarithms
Exponentials
Logarithms
Product Rule
Product Rule
Quotient Rule
Quotient Rule
Power Rule
Power Rule
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Common Logarithms

• Logarithms to the base 10 are used extensively in
• science and engineering and are called common
  logs and are often written in short form
  notation with the base 10 omitted as
• which is interpreted as

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Log Problem - Example 1
The Richter scale is used to measure the
intensity of earthquakes, and is given by the
formula R 0.667(log E 4.4), where E
is the energy released in an earthquake, measured
in joules. The San Francisco earthquake of 1906
registered 8.2 on the Richter scale, and one in
1989 measured 7.1. What is the percentage of
energy released in the 1989 earthquake compared
to the one in 1906?
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Log Problem - Example 2

• The decibel level of sound from a stereo set
  decreases with distance according to the
  relationship
• dB 10 log(320x107 / r2)
• Determine the decibel rating at 5, 10, and 15ft.
  Express the relationship in the form
• dB a b log r

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Natural Logarithms (I)

• Many engineering dynamic systems can be modeled
  by an exponential equation (ex f(x) ) where
  e is the number
• e 2.718281826
• Such systems include chemical reactions,
  materials growth, electrical circuits and the
  decay of radiation.
• Consider

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Natural Logarithms (II)

• Because they are so prevalent, natural logs are
  usually written in short form as
• Instead of
• To convert from common to natural logs

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Log Problem - Example 3

• The equation for a chemical reaction, R, to reach
  a certain state as a function of time, t (yrs),
  is given by
• How long until the reaction reaches 50 of
  its final value?

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Log Problem - Example 4

• Stronium-90, a nuclear waste, decays
  exponentially according to
• where initial mass, r rate of decay
  (1/yrs) and t time (yrs)
• How long will it take 1 kg of Stronium-90 to
  decay to 800grams at an annual rate of decay of
  2.48 ?