Logarithms Worksheets by css39027

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									   Algebra 2


Section 10-4 Common Logarithms
Section 10-5 Natural Logarithms
What You'll Learn
Why It's Important

    • To identify the characteristic and the
      mantissa of a logarithm
    • To find common logarithms and
      antilogarithms
    • To find natural logarithms of numbers
    • You can use common logarithms to
      solve problems involving astronomy
      and acoustics.
    • You can use natural logarithms to solve
      problems involving sales and physics
Common Logarithms

   • One of the more useful logarithms
     is base 10, because our number
     system is base10
   • Base 10 logarithms are called
     common logarithms
   • These are usually written without
     subscript 10,
      – so log10x is written as log x
What is the relationship?

    • Find the logarithms of 5.94, 59.4, 594 and
      5940
       – What do you notice?
What is the relationship?

    • Find the logarithms of 5.94, 59.4, 594 and
      5940
       – What do you notice?




       – The decimal parts are the same, the
         integer parts are different
Mantissa & Characteristic


    • The decimal part is called the mantissa
       – The logarithm of a number between 1
         and 10
    • The integer part is called the
      characteristic
       – The integer used to express a base 10
         logarithm as the sum of an integer and
         a positive decimal
       – The characteristic is the exponent of
         10 when the original number is
         expressed in scientific notation.
Example 1

   • Use a scientific or graphing calculator to
     find the logarithm for each number
     rounded to four decimal places. Then
     state the mantissa and characteristic.

   • A. log 120
                                       Reminder of
                                        what a log
                                         really is
   • B. log 0.12
Solution Example 1A

   • Use a scientific or graphing calculator to
     find the logarithm for each number rounded
     to four decimal places. Then state the     The characteristic is the
                                                exponent of 10 when
     mantissa and characteristic.               the original number is
                                                expressed in scientific
   • A. log 120 = 2.0792                        notation.



                             Mantissa = .0792
                             Characteristic = 2      120 = 1.20 x 102


   • You can set your calculator to round to 4
     decimal places
Solution Example 1B

   • Use a scientific or graphing calculator to find
     the logarithm for each number rounded to four
     decimal places. Then state the mantissa and
     characteristic.
   • B. log 0.12 = -0.9208
                        Mantissa = .0792
                        Characteristic = -1
                                              0.12 = 1.2 x 10-1
   • The mantissa is usually expressed as a
     positive number. To avoid negative
     mantissas, we rewrite the negative mantissa
     as the difference of a positive number and an
     integer, usually 10
Example 2

   • Use a calculator to find the logarithm
     for 0.0038 rounded to four decimal
     places. Then state the mantissa and
     characteristic.
Solution Example 2

   • Use a calculator to find the logarithm for
     0.0038 rounded to four decimal places. Then
     state the mantissa and characteristic.


                    Log 0.0038 = -2.4202

      Mantissa = 0.5798
      Characteristic = -3

                    0.0038 = 3.8 x 10-3
Antilogarithm

   • Sometimes an application of logarithms
     requires that you use the inverse of
     logarithms, exponentiation.
   • When you are given the logarithm of a
     number and asked to find the number,
     you are finding the antilogarithm.
      – That is, if log x = a, then x = antilog a
Example 3

   • Use a calculator to find the
     antilogarithm of 3.073
Solution Example 3

   • Use a calculator to find the antilogarithm
     of 3.073




       ≈1183

        Check:
Natural Logarithms

   • The number e used in the exponential
     growth problem on page 622 is used
     extensively in science and
     mathematics
   • It is an irrational number whose value
     is approximately 2.718.
   • e is the base for the natural
     logarithms, which are abbreviated ln
   • The natural logarithm of e is 1
   • All properties of logarithms that we
     have learned apply to the natural
     logarithms as well
   • The key marked         on your calculator
     is the natural logarithm key
Example 4

   • Use a calculator to find ln 3.925
Solution Example 4

   • Use a calculator to find ln 3.925




   • ≈ 1.3674
Antilogarithms

   • You can take antilogarithms of
     natural logarithms as well.
   • The symbol for the antilogarithm of x
     is antiln x
Example 5

   • A. Find x if ln x ≈3.4825

   • B. find e if ln e = 1
Solution Example 5A

   • A. Find x if ln x ≈3.4825
   • x ≈ antiln 3.4825




   • x ≈ 32.5410
Solution Example 5B

   • B. find e if ln e = 1
   • e = antiln 1




   • e ≈ 2.7183

								
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