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Homework Help: Logarithm state the value of x for which the equation is defined

  1. May 11, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider the equation:
    3logx5+2logx2-log1/x2=3
    a)State which values of x for which the equation is defined.
    b)Solve the equation for x.
    2. Relevant equations


    3. The attempt at a solution
    3logx5+2logx2-log1/x2=3
    =logx53+logx22-log1/x2=3
    =logx125+logx4-log1/x2=3
    =logx500-log1/x2=3

    =logx2=logx2/logx(1/x)change of base
    =-logx2

    =logx125+logx4+logx2=3
    =logx500+logx2=3
    =logx1000=3
    =x3=1000
    x=10

    Please check if you don't mind?
     
    Last edited: May 11, 2015
  2. jcsd
  3. May 11, 2015 #2

    SammyS

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    Not too difficult to check.

    3logx5+2logx2-log1/x2=3

    becomes:
    3log105+2log102-log1/102=3 .​

    See if the following is true.
    ##\displaystyle\ 10^{\displaystyle\left(3\log_{10}(5)+2\log_{10}(2)-\log_{1/10}(2)\right)}=10^3\ ##​
     
  4. May 11, 2015 #3
    Yes it works.
    I dont understand what 9.1 wants me to do?
     
    Last edited: May 11, 2015
  5. May 11, 2015 #4

    SammyS

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    What's 9.1 ?
     
  6. May 11, 2015 #5
    State which values of x for which the equation is defined.
     
  7. May 11, 2015 #6

    Mark44

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    For what values of x does logx(something) make sense? Same question for log1/x(something).
     
  8. May 11, 2015 #7

    SammyS

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    Several ways to figure this out.

    1. From definition of the logarithm.
    What does it mean, particularly for base, b, if logb(A) = C ?​

    2. From change of base, along with knowing the domain of the logarithm function.
    What are logx(A) and log1/x(A) ?​
    ...
     
  9. May 11, 2015 #8
    logb(A) = C
    bC=A
    x>1
     
  10. May 11, 2015 #9

    SammyS

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    Doesn't that mean that 1/x < 1 ?
     
  11. May 11, 2015 #10
  12. May 11, 2015 #11

    BvU

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    Still lets 1/x < 1 ! But: is that a problem ?

    Try to put SammyS' question 1 in words.
     
  13. May 11, 2015 #12

    SammyS

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    What are the restrictions on the base, b, if C and A are to be real numbers?

    I think the change of base route might lead to the answer more quickly, but if you're to understand logarithmic & exponential functions, then eventually you need to confront this issue regarding the base.
     
  14. May 11, 2015 #13
    a logarithm is undefined when x<0
    1/x will be 1/3 but still x>0 so the logarithm will be defined.
    x>0
     
  15. May 11, 2015 #14

    SammyS

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    I don't see what 1/3 has to do with anything here.

    Yes, as you state, "a logarithm is undefined when x<0." .... and, yes, "x > 0" .

    Now, are you referring to the base of the logarithm ?
     
  16. May 11, 2015 #15
    Yes, the base of log cannot be smaller than 0
     
  17. May 11, 2015 #16

    SammyS

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    Can it be zero ?

    Why or why not?
     
  18. May 11, 2015 #17
    It is not defined, cannot be calculated, gives an error.
    Its a rule
     
  19. May 11, 2015 #18

    SammyS

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    What is not defined?
     
  20. May 11, 2015 #19
    the logarithm
     
  21. May 11, 2015 #20

    SammyS

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    Please make a complete statement.

    What logarithm is not defined?
     
  22. May 11, 2015 #21
    The real logarithmic function logb(x) is defined only for x>0. So the base b logarithm of zero is not defined.
     
  23. May 11, 2015 #22

    SammyS

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    This is precisely why we needed you to make a full statement here.

    The question (in Post #1) refers to x.

    In this problem, x appears, not as the argument of the logarithm, but appears as the base, as well as 1/x appears as the base.
     
  24. May 11, 2015 #23
    Thank you so much for all your help.
    Have a great night.
    Jaco
     
  25. May 11, 2015 #24

    Mark44

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    Is the log function logb(x) defined if b = 1?

    Edit: Fixed my bonehead typo of "if x = 1"
     
    Last edited: May 15, 2015
  26. May 14, 2015 #25
    i have tried to follow the steps i am also having a problem in stating the values of x for which the equation is defined
     
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