Math-LOGS

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Express x in terms of a,b and c.
log x = 1/2 (log a + log b - log c)
Please solve and explain how to do this type of problem, thank you!

  • Math-LOGS -

    duplicate

  • Math-LOGS -

    To solve
    log x = 1/2 (log a + log b - log c)
    we need to know some properties of logarithms, namely:

    1. log(a)+log(b) = log(ab)
    2. (1/2)log(a)=log(a-1/2)=log(√(a))
    3. eln x = x, or
    10log10 x = x

    Proceeding to simplify the right-hand-side,
    log x = 1/2 (log a + log b - log c)
    = (1/2)log a + (1/2)log b + (1/2)log c
    = log(√a) + log(√b) + log(√c)
    = log(√a √b √c)
    = log(√(abc))

    Assuming log() stands for logarithm to the base e,
    elog x = elog(√(abc))
    x = √(abc)

  • correction -

    2nd rule of logarithm should read:
    2. (1/2)log(a)=log(a1/2)=log(√(a))

    and the solution has to be corrected because of an erroneous sign:

    Proceeding to simplify the right-hand-side,
    log x = 1/2 (log a + log b - log c)
    = (1/2)log a + (1/2)log b - (1/2)log c
    = log(√a) + log(√b) - log(√c)
    = log(√a √b / √c)
    = log(√(ab/c))

    Assuming log() stands for logarithm to the base e,
    elog x = elog(√(ab/c))
    x = √(ab/c)

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