add log of different bases

Discussion in 'Undergraduate Math' started by darklord, Jan 15, 2005.

  1. darklord

    darklord Guest

    Hi,

    I have a question regarding logarithm. I know how to add and subtract
    log of a same base, however I have this problem where I must add two
    logs of different bases. Here is the problem, which I am supposed to
    find x without using calculator.

    log(8)4 - log(8)X = log(8)5 + log(16)6

    The logarithm of base 16 throws me off. My guess is that I probably
    can break log(16)6 into two log(8) but I don't know whether the 6
    would still stay the same? Anyways, could you please show me how to
    solve this problem? Thanks
     
    darklord, Jan 15, 2005
    #1
    1. Advertisements

  2. There is a "change of base" formula for logarithms.

    log(a) x = ( log(b) x ) / ( log(b) a )

    where a and b are positive numbers, not equal to 1. Using this, you
    can rewrite log(16) 6 as

    log(16) 6 = ( log(8) 6 ) / ( log(8) 16 )

    Also, you might notice that 8^(4/3) = 16, so log(8) 16 = 4/3. Thus,

    ( log(8) 6 ) / ( log(8) 16 ) = ( log(8) 6 ) / ( 4/3 )

    = ( 3/4 ) ( log(8) 6 )

    = log(8) 6^(3/4)

    Hope this helps,

    Brian
     
    Brian VanPelt, Jan 15, 2005
    #2
    1. Advertisements

  3. Hint


    Look through your Book or notes for the formula for convering logs
    from one base to another base, then use that to change base 16 to
    base 8


    Patrick
     
    Patrick FitzGerald, Jan 15, 2005
    #3
  4. darklord

    Stan Brown Guest

    Log_16(x) = log_8(x) / log_8(16)

    Since 8^(4/3) = 16, log_8(16) = 4/3

    Therefore log_16(6) = log_8(6) / (4/3) = (3/4) * log_8(6) =
    log_8(6^(3/4))
     
    Stan Brown, Jan 15, 2005
    #4
    1. Advertisements

Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments (here). After that, you can post your question and our members will help you out.