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									 7-4 Properties of Logarithms
7-4 Propertiesof Logarithms




                          Warm Up
                          Lesson Presentation
                          Lesson Quiz




 Holt Algebra 2
Holt McDougal Algebra 2
7-4 Properties of Logarithms

   Warm Up
    Simplify.
    1. (26)(28)           214   2. (3–2)(35) 33


    3.                    38    4.           44


     5. (73)5             715
   Write in exponential form.

    6. logx x = 1 x1 = x        7. 0 = logx1 x0 = 1

Holt McDougal Algebra 2
7-4 Properties of Logarithms

                          Objectives
  Use properties to simplify logarithmic
  expressions.
  Translate between logarithms in any
  base.




Holt McDougal Algebra 2
7-4 Properties of Logarithms


     The logarithmic function for pH that you saw in
     the previous lessons, pH =–log[H+], can also be
     expressed in exponential form, as 10–pH = [H+].


     Because logarithms are exponents, you can derive
     the properties of logarithms from the properties of
     exponents




Holt McDougal Algebra 2
7-4 Properties of Logarithms

  Remember that to multiply
  powers with the same base,
  you add exponents.




Holt McDougal Algebra 2
7-4 Properties of Logarithms


  The property in the previous slide can be used in
  reverse to write a sum of logarithms (exponents)
  as a single logarithm, which can often be
  simplified.


  Helpful Hint
  Think: logj + loga + logm = logjam




Holt McDougal Algebra 2
7-4 Properties of Logarithms
                 Example 1: Adding Logarithms

  Express log64 + log69 as a single logarithm.
  Simplify.

      log64 + log69

                            To add the logarithms, multiply
      log6 (4  9)
                            the numbers.

      log6 36               Simplify.


      2                     Think: 6? = 36.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
                     Check It Out! Example 1a


Express as a single logarithm. Simplify, if possible.


      log5625 + log525


      log5 (625 • 25)           To add the logarithms, multiply
                                the numbers.
      log5 15,625               Simplify.

      6                         Think: 5? = 15625
Holt McDougal Algebra 2
7-4 Properties of Logarithms
                     Check It Out! Example 1b


Express as a single logarithm. Simplify, if possible.


      log 1 27 + log 1            1
           3                  3   9

                          1
      log 1 (27 •             )       To add the logarithms, multiply
           3              9
                                      the numbers.
      log 1 3                         Simplify.
           3


                                               1 ?
      –1                              Think:   3 =   3
Holt McDougal Algebra 2
7-4 Properties of Logarithms


   Remember that to divide
   powers with the same base,
   you subtract exponents


   Because logarithms are exponents, subtracting
   logarithms with the same base is the same as
   finding the logarithms of the quotient with that
   base.




Holt McDougal Algebra 2
7-4 Properties of Logarithms




     The property above can also be used in reverse.

     Caution
      Just as a5b3 cannot be simplified, logarithms
      must have the same base to be simplified.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
             Example 2: Subtracting Logarithms

  Express log5100 – log54 as a single logarithm.
  Simplify, if possible.


      log5100 – log54

                          To subtract the logarithms,
      log5(100 ÷ 4)
                          divide the numbers.

      log525              Simplify.


       2                  Think: 5? = 25.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
                      Check It Out! Example 2

  Express log749 – log77 as a single logarithm.
  Simplify, if possible.

      log749 – log77

                           To subtract the logarithms,
      log7(49 ÷ 7)
                           divide the numbers

      log77                Simplify.

      1                   Think: 7? = 7.

Holt McDougal Algebra 2
7-4 Properties of Logarithms


      Because you can multiply logarithms, you can
      also take powers of logarithms.




Holt McDougal Algebra 2
7-4 Properties of Logarithms
 Example 3: Simplifying Logarithms with Exponents


 Express as a product. Simplify, if possible.


  A. log2326                  B. log8420


      6log232                 20log84
                Because                           Because
                                                    2
      6(5) = 30 25 = 32,      20(   2
                                    3
                                        )=   40
                                             3
                                                  8 = 4,
                                                    3

                                                          2
                log232 = 5.                       log84 = 3 .



Holt McDougal Algebra 2
7-4 Properties of Logarithms
                      Check It Out! Example 3

 Express as a product. Simplify, if possibly.


  a. log104                         b. log5252


      4log10                        2log525
                      Because                    Because
      4(1) = 4        101 = 10,     2(2) = 4     52 = 25,
                      log 10 = 1.                log525 = 2.


Holt McDougal Algebra 2
7-4 Properties of Logarithms
                         Check It Out! Example 3


 Express as a product. Simplify, if possibly.


  c. log2 (     1
                2
                    )5

                1
      5log2 (   2   )
                           Because
                                  1
      5(–1) = –5           2–1 = 2 ,
                                1
                           log2 2 = –1.



Holt McDougal Algebra 2
7-4 Properties of Logarithms

   Exponential and logarithmic operations undo each
   other since they are inverse operations.




Holt McDougal Algebra 2
7-4 Properties of Logarithms
               Example 4: Recognizing Inverses


 Simplify each expression.


 a. log3311               b. log381             c. 5log 10  5




     log3311                log33  3  3  3     5log 10
                                                        5



                                 4
     11                     log33                  10
                            4




Holt McDougal Algebra 2
7-4 Properties of Logarithms
                      Check It Out! Example 4


 a. Simplify log100.9                b. Simplify 2log (8x)
                                                     2




      log 100.9                         2log (8x)
                                             2




      0.9                               8x




Holt McDougal Algebra 2
7-4 Properties of Logarithms

    Most calculators calculate logarithms only in base
    10 or base e (see Lesson 7-6). You can change a
    logarithm in one base to a logarithm in another
    base with the following formula.




Holt McDougal Algebra 2
7-4 Properties of Logarithms
     Example 5: Changing the Base of a Logarithm


     Evaluate log328.

    Method 1 Change to base 10

                    log8
     log328 =
                   log32

                0.903
              ≈            Use a calculator.
                 1.51

               ≈ 0.6       Divide.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
                          Example 5 Continued


     Evaluate log328.

    Method 2 Change to base 2, because both 32
             and 8 are powers of 2.


     log328 =
                    log28   3
                          =        Use a calculator.
                   log232   5


              = 0.6

Holt McDougal Algebra 2
7-4 Properties of Logarithms
                     Check It Out! Example 5a


     Evaluate log927.

    Method 1 Change to base 10.

                      log27
     log927 =
                      log9

                1.431
              ≈               Use a calculator.
                0.954

               ≈ 1.5          Divide.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
            Check It Out! Example 5a Continued


    Evaluate log927.

    Method 2 Change to base 3, because both 27
             and 9 are powers of 3.


     log927 =
                    log327  3
                          =     Use a calculator.
                    log39   2


              = 1.5

Holt McDougal Algebra 2
7-4 Properties of Logarithms
                     Check It Out! Example 5b


     Evaluate log816.

    Method 1 Change to base 10.

                      log16
     Log816 =
                      log8

                1.204
              ≈               Use a calculator.
                0.903

               ≈ 1.3          Divide.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
            Check It Out! Example 5b Continued


    Evaluate log816.

    Method 2 Change to base 4, because both 16
             and 8 are powers of 2.


     log816 =
                     log416      2
                              =
                     log48      1.5 Use a calculator.

              = 1.3



Holt McDougal Algebra 2
7-4 Properties of Logarithms

      Logarithmic scales are useful for measuring
      quantities that have a very wide range of
      values, such as the intensity (loudness) of a
      sound or the energy released by an
      earthquake.

       Helpful Hint
       The Richter scale is logarithmic, so an increase of
       1 corresponds to a release of 10 times as much
       energy.




Holt McDougal Algebra 2
7-4 Properties of Logarithms
                Example 6: Geology Application

  The tsunami that devastated parts of Asia in
  December 2004 was spawned by an
  earthquake with magnitude 9.3 How many
  times as much energy did this earthquake
  release compared to the 6.9-magnitude
  earthquake that struck San Francisco in1989?

                              The Richter magnitude of an
                              earthquake, M, is related to the
                              energy released in ergs E given
                              by the formula.
                               Substitute 9.3 for M.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
                          Example 6 Continued


                                                            3
                                   Multiply both sides by   2   .

                     E           Simplify.
        13.95 = log  11.8 
                    10 

                                   Apply the Quotient Property
                                   of Logarithms.
                                   Apply the Inverse Properties of
                                   Logarithms and Exponents.


Holt McDougal Algebra 2
7-4 Properties of Logarithms
                          Example 6 Continued


                              Given the definition of a logarithm,
                              the logarithm is the exponent.



                              Use a calculator to evaluate.



   The magnitude of the tsunami was 5.6  1025 ergs.


Holt McDougal Algebra 2
7-4 Properties of Logarithms
                          Example 6 Continued




                                   Substitute 6.9 for M.

                                                            3
                                   Multiply both sides by   2   .


                                   Simplify.

                                   Apply the Quotient Property
                                   of Logarithms.
Holt McDougal Algebra 2
7-4 Properties of Logarithms
                          Example 6 Continued

                          Apply the Inverse Properties of Logarithms
                          and Exponents.

                           Given the definition of a logarithm,
                           the logarithm is the exponent.

                           Use a calculator to evaluate.
      The magnitude of the San Francisco earthquake
      was 1.4  1022 ergs.
      The tsunami released            5.6  1025 = 4000 times as
                                      1.4  1022
      much energy as the earthquake in San Francisco.
Holt McDougal Algebra 2
7-4 Properties of Logarithms
                      Check It Out! Example 6

  How many times as much energy is released
  by an earthquake with magnitude of 9.2 by an
  earthquake with a magnitude of 8?




                                 Substitute 9.2 for M.

                                                          3
                                 Multiply both sides by   2   .


                                 Simplify.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
             Check It Out! Example 6 Continued

                          Apply the Quotient Property
                          of Logarithms.

                          Apply the Inverse Properties of
                          Logarithms and Exponents.

                          Given the definition of a logarithm,
                          the logarithm is the exponent.


                          Use a calculator to evaluate.

The magnitude of the earthquake is 4.0  1025 ergs.
Holt McDougal Algebra 2
7-4 Properties of Logarithms
             Check It Out! Example 6 Continued




                             Substitute 8.0 for M.


                                                      3
                             Multiply both sides by   2   .


                             Simplify.


Holt McDougal Algebra 2
7-4 Properties of Logarithms
             Check It Out! Example 6 Continued


                          Apply the Quotient Property
                          of Logarithms.

                          Apply the Inverse Properties
                          of Logarithms and Exponents.

                          Given the definition of a
                          logarithm, the logarithm is the
                          exponent.


                           Use a calculator to evaluate.
Holt McDougal Algebra 2
7-4 Properties of Logarithms
             Check It Out! Example 6 Continued


      The magnitude of the second earthquake was
      6.3  1023 ergs.

       The earthquake with a magnitude 9.2 released
       was     4.0  1025 ≈ 63 times greater.
               6.3  1023




Holt McDougal Algebra 2
7-4 Properties of Logarithms
                  Lesson Quiz: Part I
        Express each as a single logarithm.
        1. log69 + log624        log6216 = 3

        2. log3108 – log34       log327 = 3

        Simplify.

        3. log2810,000           30,000

        4. log44x –1             x–1

        5. 10log125              125
        6. log64128              7
                                 6
Holt McDougal Algebra 2
7-4 Properties of Logarithms
                          Lesson Quiz: Part II

      Use a calculator to find each logarithm to
      the nearest thousandth.
      7. log320                   2.727

      8. log 1 10                 –3.322
               2

      9. How many times as much energy is released by
         a magnitude-8.5 earthquake as a magntitude-
         6.5 earthquake?
         1000



Holt McDougal Algebra 2

								
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