Results 1 to 3 of 3

Math Help - Log Law Intuition

  1. #1
    Newbie
    Joined
    May 2011
    Posts
    13

    Log Law Intuition

    What is the intuition for this logarithm law?

    logbx = (logax) / (logab)



    There is intuition for other log laws. For example:

    log(100) + log (1000)
    = 2 factors of 10 + 3 factors of 10
    = 5 factors of 10
    = 5
    = log (100x1000)
    = (2 + 3) factors of 10
    = 5 factors of 10
    = 5

    So more generally, log(a) + log(b) = log(ab). We're just counting the factors of 10 in a and b, so it doesn't matter whether we count them separately or together. It's equal either way.


    Is there intuition for the equation in red above? Although it's trivial to prove, seeing the equation (and using it) in isolation seems more mechanical (e.g. plugging numbers into the Pythagorean Theorem because I see a right triangle) than logical (e.g. solving x + 5 = 24 by subtracting 5 from both sides).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,969
    Thanks
    1009
    \displaystyle \begin{align*}x &= b^y \\ \log_b{x} &= y \end{align*}

    Also

    \displaystyle \begin{align*} x &= b^y \\ \log_a{x} &= \log_a{\left(b^y\right)} \\ \log_a{x} &= y\log_a{b} \\ \frac{\log_a{x}}{\log_a{b}} &= y\end{align*}


    Therefore \displaystyle \log_b{x} = \frac{\log_a{x}}{\log_a{b}}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    I originally wrote just what Prove It did (only slower!). However, you seem to be using "intuition" to mean something different from a regular proof.

    Perhaps this is the kind of thing you mean:
    u= log_a(x) means that x has u factors of a. v= log_b(x) means x has v factors of b. w= log_b(a) means that a has w factors of b. Well, if x has u factors of a and each a has w factors of b, then x has uw factors of b.

    That is, v= uw so u= v/w which translates as
    log_a(x)= \frac{log_b(x)}{log_b(a)}
    Last edited by HallsofIvy; May 27th 2011 at 08:43 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Intuition for inclusion exclusion
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: October 9th 2011, 06:28 AM
  2. Compound Interest - Intuition is wrong
    Posted in the Business Math Forum
    Replies: 4
    Last Post: April 12th 2011, 07:49 PM
  3. Replies: 3
    Last Post: August 16th 2010, 12:32 AM

Search Tags


/mathhelpforum @mathhelpforum