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Math Help - Log proofs

  1. #1
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    Log proofs

    Hi I am terrible at proofs. I am going to write 'log base b' as logb. I need to prove the following:
    1. logb 1=0
    2. logb b=1
    3. logb 1/b=-1
    4. logb 1/x=-logb x

    I realise this may a bit time consuming so thank you in advance. Or if anyone has seen these proofs somewhere? Thanks.
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  2. #2
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    Quote Originally Posted by slaypullingcat View Post
    Hi I am terrible at proofs. I am going to write 'log base b' as logb. I need to prove the following:
    1. logb 1=0
    2. logb b=1
    3. logb 1/b=-1
    4. logb 1/x=-logb x

    I realise this may a bit time consuming so thank you in advance. Or if anyone has seen these proofs somewhere? Thanks.
    In each case you should use the fact that \log_b a = c \iff b^c = a.
    Last edited by mr fantastic; May 12th 2009 at 04:43 AM.
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  3. #3
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    Quote Originally Posted by slaypullingcat View Post
    Hi I am terrible at proofs. I am going to write 'log base b' as logb. I need to prove the following:
    1. logb 1=0
    2. logb b=1
    3. logb 1/b=-1
    4. logb 1/x=-logb x

    I realise this may a bit time consuming so thank you in advance. Or if anyone has seen these proofs somewhere? Thanks.
    These all follow, fairly immediately, from the basic definition of logarithms, as pointed out in the previous reply. Where are you having difficulty?

    Please be specific. Thank you!
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  4. #4
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    I'm having trouble with the fourth one. I guess I can show the relationships as stated in the link you posted, but, I don't feel like I have proved anything.

    For example, with No.3, I just wrote
    log b (1/b)=-1 then log 4 (1/4)=-1 because 4^-1=1/4
    Thats it, I did the same for the first two but I'm stuck on the fourth. I just don't know if what I have done constitutes a proof?
    Thank you
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  5. #5
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    Quote Originally Posted by slaypullingcat View Post
    I'm having trouble with the fourth one. I guess I can show the relationships as stated in the link you posted, but, I don't feel like I have proved anything.

    For example, with No.3, I just wrote
    log b (1/b)=-1 then log 4 (1/4)=-1 because 4^-1=1/4
    Thats it, I did the same for the first two but I'm stuck on the fourth. I just don't know if what I have done constitutes a proof?
    Thank you
    \log_b \frac{1}{x} = c .... (1)

    \Rightarrow b^c = \frac{1}{x} \Rightarrow b^{-c} = x

    \Rightarrow -c = \log_b x \Rightarrow c = - \log_b x .... (2)

    Now equate (1) and (2).
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