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Expand using logarithm properties: log6 x(x+5)/x^6

I am so confused about this problem. Could anyone explain this to me?

  • Math -

    just as powers of products add (6^2 * 6^3 = 6^5) and subtract (6^8 / 6^3 = 6^5)

    log of a product is the sum of the logs, and log of a quotient is difference of logs. That's because of the definition of log:

    log6N is the power of 6 needed to get N. Even though it looks weird, a log is just an exponent!

    So, log x(x+5)/x^6
    = logx + log(x+5) - log x^6

    Now, log x^6 = log x*x*x*x*x*x = logx + logx + logx + logx + logx = 6logx

    and now qwe end up with

    logx + log(x+5) - 6logx
    = log(x+5) - 5logx

    That could also have been achieved by noticing that

    x(x+5)/x^6 = (x+5)/x^5
    notice how the power subtracted when factoring out the x?

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