Math: Logarithms

(log(x^6)+2log(y^3))/log(xy)

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  1. remember a*log(b)= log(b^a)
    and log a+ log b= log (ab)

    and log a/logb= log (a)-log (b)

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    bobpursley
  2. recall that when the expression inside the log is raised to a certain power, we can write the exponent outside of the log, for instance:
    log (x^2) = 2 log x
    log [(2y)^0.5] = 0.5 log 2y
    thus for the problem, we can rewrite it as:
    (log(x^6) + 2log(y^3)) / log(xy)
    (6 log x + 6 log y ) / log (xy)
    6 (log x + log y) / log (xy)
    now recall that the sum of log with the same base is the same as the log of the product of the expressions inside the parenthesis, for instance:
    log 2 + log x = log 2x
    log a + log b = log ab
    thus we can further simplify this as:
    6 (log x + log y) / log (xy)
    6 (log xy) / log xy
    cancelling out the log terms, we have 6

    hope this helps~ :)

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