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Math Help - Log from earlier question, just need confirmation!

  1. #1
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    Log from earlier question, just need confirmation!

    I have log4(x^2 - 9) - log4(x + 3) = 3

    I get 4^3 = (x^2 - 9) / (x + 3 )

    and then 64 = (x - 3)

    x = 67

    Correct?
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  2. #2
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    Correct! . . . Nice work!

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  3. #3
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    Thanks! I do have another one here I am working on and could use a jump start. Not sure how to write it. . .

    Write the expression 21log3(cube root of x) + log 3(9x^2) - log 3(9) as a single logarithm....
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  4. #4
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    Quote Originally Posted by rtwilton View Post
    I do have another one here I am working on and could use a jump start. Not sure how to write it. . .

    Write the expression 21log3(cube root of x) + log 3(9x^2) - log 3(9) as a single logarithm....
    Use these formulae :
    a \log b =\log b^a
    \log a + \log b- \log c=\log \left(\frac{ab}{c}\right)

    and this holds true for any base of the logarithm.

    note that \sqrt[3]{x}=x^{1/3}


    So 21 \log_3 \sqrt[3]{x}+\log_3 (9x^2)-\log_3(9)=\dots=\log_3 \left(\frac{x^7 \cdot 9x^2}{9}\right)=\log_3 (x^9)=9 \log_3(x)

    d'you get all the calculations ?
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  5. #5
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    I could really use the steps in between. . .I am not making the leap, sorry! I know that subtraction is division, addition is multiplication, correct??
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  6. #6
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    Quote Originally Posted by rtwilton View Post
    I could really use the steps in between. . .I am not making the leap, sorry! I know that subtraction is division, addition is multiplication, correct??
    Yep !

    But don't make this mistake :
    \log(a+b) {\color{red} \neq } \log(a) \cdot \log(b)

    Okay, here are the steps, hope you'll know how to do further exercises later

    ~~~~~~~~~~~~~~~~~~~~~~~~~~
    21 \log_3 (\sqrt[3]{x})=21 \log_3 (x^{1/3})

    from the property a log(b)=log(b^a), this is :
    \log_3 \left((x^{1/3})^{21}\right)

    use the rule (a^b)^c=(a^c)^b=a^{bc} and you'll get :

    21 \log_3 (\sqrt[3]{x})=\log_3 \left(x^{21/3}\right)=\log_3 (x^7)


    this is the main step of the leap


    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    I've just seen a better way to do it :

    21 \log_3 (x^{1/3})+\log_3(9x^2)-\log_3(9)=7 \log_3(x)+\log_3(9)+\log_3(x^2)-\log_3(9) =7\log_3(x)+2 \log_3(x)=9 \log_3(x)
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  7. #7
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    Thank you so much, I can see it now. Yes, I believe I will be able to work through future problems, I hope!!
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