§4.4 Common and Natural Logarithms by pharmphresh26

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									        §4.4 Common and Natural Logarithms

 Common and Natural Logarithms
 Common logarithms are logarithms with base 10, and denoted by log.
Natural logarithms are logarithms with base e, and denoted by ln.

    Logarithm Notation
                log x = log10 x          Common logarithm
                   ln x = loge x         Natural logarithm




                   y
               2                                               y = ln x
               1                                               y = log x
                                                                   x
                             2       4        6     8         10
             -1

             -2




                                     Figure 1


    Logarithm-Exponential Relationships
                       log x = y   is equivalent to x = 10y
                        ln x = y   is equivalent to x = ey


  Example 1. Use a calculator to evaluate each to six decimal places:
  (A). log 3184 (B). ln 0.000349
Proof. (A). log 3184 = 3.502973 (Verify: 103.52973 = 3183.99957)
  (B). ln 0.000349 = −7.960439 (Verify: 10−7.960439 = 0.000349)
  Example 2. Use a calculator to evaluate each to six decimal places:
        log 2            2
  (A).         (B). log        (C). log 2 − log 1.1
       log 1.1          1.1
Proof. log 2 = 0.30103, log 1.1 = 0.0413927.
                                          1
2


           log 2      0.30103
    (A).          =            = 7.273
          log 1.1    0.0413927
               2
    (B). log      = log 1.81818 = 0.260
              1.1
    (C). log 2 − log 1.1 = 0.30103 − 0.0413927 = 0.260


    Example Simplify using the properties of logarithmic functions.
    (A). ln 1 (B). log 10 (C). ln e2 x+1 (D). 10log 7

Solution. (A). ln 1 = 0          (B). log 10 = 1
(C). ln e2 x+1 = (2 x + 1) ln e = 1    (D). 10log 7 = 7.

                        1.10
    Example If ln 3 = √ and ln 7 = 1.95, find:
            7
    (A). ln 3 , (B). ln 3 21.

Solution. (A). ln 7 = ln 7 − ln 3 = 1.95 − 1.10 = 0.85
        √         3
(B). ln 3 21 = ln(21)1/3 = 1 ln(3 × 7) = 3 (ln 3 + ln 7) = 1 (1.10 + 1.95) =
                             3
                                            1
                                                           3
1.02.

    Applications
    Sound Intensity
    The sound intensity can be measure by decibel as following:
                                     I
                        D = 10 log        Decibel scale
                                     I0
where
  (1) D: The decibel level of the sound
  (2) I: The intensity of the sound measured in watts per square meter
      (W/m2 )
  (3) I0 : The intensity of the least audible sound that an average healthy
      young person can hear (standardized to be I0 = 10−12 watt per
      square meter).

    Sound intensity, W/m2           Sound            Decibel level
         1.0 × 10 −12        Threshold of hearing          0
         5.2 × 10−10               Whisper               27.16
          3.2 × 10−6         Normal conversation         65.05
          8.5 × 10−4             Heavy traffic             89.29
          3.2 × 10 −3            Jackhammer              95.05
           1.0 × 100          Threshold of pain           120
           8.3 × 102      Jet plane with afterburner    149.19
              Table 1. Typical Sound Intensities and Decibel Level


    Earthquake Intensity
                                                                             3


  The energy released by the earthquake can be represented by the mag-
nitude M on the Richter scale as following:
                            2     E
                      M=      log         Richter scale
                            3     E0
where
  (1) M : The Magnitude on Richter scale

   (2) E: The energy released by the earthquake, measure in joules

   (3) E0 : The energy released by a very small reference earthquake (stan-
       dardized to be E0 = 104.40 joules)

              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
                         Table 2. The Richter Scale


   Example 6. If the energy release of one earthquake is 1000 times that
of another, how much larger is the Richter scale reading of the larger than
the smaller?
Solution. Let E1 and E2 the energy releases of smaller and larger earthquake
respectively. Let
                                     2     E1
                               M1 = log
                                     3     E0
and
                                     2     E2
                               M2 = log
                                     3     E0
be the Richter equations for the smaller and larger earthquakes, respectively.
Substituting E2 = 1000 E1 into the second equation, we obtain
                              2     1000 E1
                      M2 =      log
                              3        E0
                              2                 E1
                            =     log 103 + log
                              3                 E0
                              2        2      E1
                            =   × 3 + log
                              3        3      E0
                            = 2 + M1
Thus, an earthquake with 1000 times the energy of another has a Richter
scale reading of 2 more than the other.
4


  Further reading: Sec. 4.4
Exercise: Sec. 4.4: 17, 19, 21, 23, 41, 42

								
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