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Math Help - Some Logarithm Questions

  1. #1
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    Some Logarithm Questions

    Here are some log questions that I haven't been able to solve. The problem that I'm having is that the base is different in some of them which is throwing me off.
    By the way, the first question has a little typo. The right side of the equation should say 2log9(x-3) instead of 2log5(x-3)



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  2. #2
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    10^{2y}=25
    Then,
    \left( 10^{2y} \right)^{1/2}=25^{1/2}=\sqrt{25}=5
    Thus,
    10^{2y\cdot 1/2}=10^y=5
    Raise both sides to (-1) thus,
    10^{-y}=5^{-1}=\frac{1}{5}
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    (\log x)(\log_5 10)=3
    Divide both sides by the base 5 logarithm,
    \log x=\frac{3}{\log_5 10}.
    Now there is a nice little rule that says,
    \log_a b=\frac{1}{\log_b a}
    Thus, you can really write,
    \log x=3\log_{10} 5=3\log 5=\log 5^3
    Thus,
    x=5^3=125
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    \log \frac{1}{2}+\log \frac{2}{3}+...+\log \frac{9}{10}
    Use the rule that the logarithm of a product is the sum of the logarithms,
    \log \left( \frac{1}{2}\cdot \frac{2}{3}\cdot ... \cdot \frac{9}{10} \right)
    Each one cancels,
    \log \frac{1}{10}=-1
    ---
    x=\log 2
    Use the rule I taught you and take reciprocals,
    \frac{1}{x}=\log_2 10=\log_2 (2\cdot 5)=\log_2 2+\log_2 5
    Thus,
    \frac{1}{x}=1+\log_2 5
    Thus,
    \log_2 5=\frac{1}{x}-1
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    2\log x=\log 2x
    Use the sum rule,
    2\log x=\log 2+\log x
    \log x=\log 2
    x=2
    ---
    \log_3 (x+9)=2\log_5 (x-3)
    \log_3 (x+9)=\log_5 (x-3)^2
    Change base,
    \log_5 (x+9)/\log_5 3=\log_5 (x-3)^2
    \log_5 (x+9)=\log_5 (x-3)^2\cdot \log _5 3
    Thus,
    x+9=(x^2-6x+9)^{\log_5 3}
    I do not think this is solvable.
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    Hello, SportfreundeKeaneKent!

    If x = \log_{10}2, then express \log_25 in terms of x

    We have: . \log_{10}2\:=\:x

    Take reciprocals: . \frac{1}{\log_{10}2} \:=\:\frac{1}{x} [1]


    The left side is: . \frac{1}{\log_{10}2} \:=\:\log_2(10) \:=\:\log_2(5\cdot2) \:=\:\log_2(5) + \log_2(2) \:=\:\log_2(5) + 1

    Then equation [1] becomes: . \log_2(5) + 1 \;=\;\frac{1}{x}

    Therefore: . \log_2(5)\;=\;\frac{1}{x} - 1\;=\;\frac{1-x}{x}

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    Quote Originally Posted by SportfreundeKeaneKent View Post
    ...The problem that I'm having is that the base is different in some of them which is throwing me off.
    Hello Sportsfreund,

    your calculator "knows" the logarithms to the base 10 (that's the log-button) and the logarithms to the bease e (that's th ln-button).

    You can transform the different logarithms by using the log or the ln-function of your calculator:

    \log_b{a}=\frac{\ln(a)}{\ln(b)}=\frac{\log(a)}{\lo  g(b)} .

    Quote Originally Posted by SportfreundeKeaneKent View Post
    By the way, the first question has a little typo. The right side of the equation should say 2log9(x-3) instead of 2log5(x-3)


    ...
    According to your correction you'll solve:

    \log_3(x+9)=2\log_9(x-3)\ ,\ x>3

    I use the above mentioned formula:

    \frac{\ln(x+9)}{\ln(3)}=\frac{2\ln(x-3)}{\ln(9)}=\frac{2\ln(x-3)}{2\ln(3)}

    Multiply by ln(3):

    \ln(x+9)=\ln(x-3). This equation has no real solution.

    I assume that there is maybe another typo.

    EB
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  8. #8
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    Alright, here's the answer to that typo question, by the way, you do change the exponent to 9:
    logbase3(x+9)=logbase3(x-3)^2/log9base3
    2logbase3(x+9)^2 - logbase3(x-3)^2=0
    logbase3(x+9)^2/(x-3)^2=0
    x^2-6x+9=x^2+18x+81
    24x=-72
    x=-3

    I know that that's hard to follow but I think it's right.
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  9. #9
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    Alright, here's the answer to that typo question, by the way, you do change the exponent to 9:
    logbase3(x+9)=logbase3(x-3)^2/log9base3
    2logbase3(x+9)^2 - logbase3(x-3)^2=0
    logbase3(x+9)^2/(x-3)^2=0
    x^2-6x+9=x^2+18x+81
    24x=-72
    x=-3

    I know that that's hard to follow but I think it's right.
    hello Sportsfreund,

    it isn't so hard to follow your calculations. You calculated correctly. Nevertheless x = -3 is not a solution of your equation:

    I copy the line where I can show you why:

    "logbase3(x+9)=logbase3(x-3)^2/log9base3"

    Now plug in x = -3. You'll get

    logbase3(-3+9)=logbase3(-3-3)^2/log9base3

    which is the same as

    logbase3(6)=logbase3(-6)^2/log9base3

    And it is impossible to calculate the logarithm of a negative number. Thus x = -3 is not a solution of your equation.

    So sorry to disappoint you.

    EB
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