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Math Help - prove (log a b^2).(log b a^3) = 6

  1. #1
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    prove (log a b^2).(log b a^3) = 6

    can anyone help with this question? cheers
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by 24680 View Post
    can anyone help with this question? cheers
    Before I start I would like to suggest that you repeat the problem statement in the written portion of your post. It would be easier for us to see what you are trying to write, particularly when we get LaTeX back.

    I presume that "log a b^2" is "log to the base a of b^2? Let me rewrite that as log(a)b^2.

    [log(a)b^2]*[log(b)a^3] = [2*log(a)b]*[3*log(b)a] = 6*[log(a)b]*[log(b)a]

    There are probably several ways to finish this. I am going to use the change of base formula to change both of these to log base 10 (which I will write simply as "log x.")

    The change of base formula is:
    log(k)x = (log x)/(log k) to convert to log base 10.

    So
    log(a)b = (log b)/(log a)
    log(b)a = (log a)/(log b)

    So
    [log(a)b^2]*[log(b)a^3] = 6*[log(a)b]*[log(b)a] = 6*(log b)/(log a)*(log a)/(log b) = 6

    -Dan
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by 24680 View Post
    can anyone help with this question? cheers
    (log_a b^2).(log_b a^3)

    ..................= [(log_b b^2)/(log_b (a))].(log b a^3)

    ..................= (log_b b^2).[(log b a^3)/(log_b (a))]

    ..................= (log_b b^2).(log a a^3)

    ..................= 2x3 = 6

    RonL
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  4. #4
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    Quote Originally Posted by 24680 View Post
    can anyone help with this question? cheers
    Hello,

    you only have to use the property: a^(log_a(b)) = b

    expand the powers:

    2*log_a(b) * 3 * log_b(a) = 6.

    6 * [log_a(b) * log_b(a)] = 6

    6 * [log_a(b^(log_b(a)))] = 6

    6 * [log_a(a)] = 6

    6 = 6


    tschüss

    EB
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