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Math Help - Laws of logarithms

  1. #1
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    Laws of logarithms

    I am familiar with the laws of logarithms, and there are three exam questions here that are each worth three marks:

    1] Show that if x > 0,
     \log_a x^k = k \log_a x

    2] Given that x > 0, y > 0, show that
     \log_a \frac{x}{y} = \log_a x - \log_a y

    3] Given that x > 0, y > 0, show that
     \log_a (xy) = \log_a x + \log_a y

    How would you prove these laws? Would you show how you reach them or possibly use an example to do this?
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  2. #2
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    Laws of Logarithms

    Hello db5vry
    Quote Originally Posted by db5vry View Post
    I am familiar with the laws of logarithms, and there are three exam questions here that are each worth three marks:

    1] Show that if x > 0,
     \log_a x^k = k \log_a x

    2] Given that x > 0, y > 0, show that
     \log_a \frac{x}{y} = \log_a x - \log_a y

    3] Given that x > 0, y > 0, show that
     \log_a (xy) = \log_a x + \log_a y

    How would you prove these laws? Would you show how you reach them or possibly use an example to do this?
    I think that you are expected to prove each of these by using the definition of a logarithm, which is:

    'The log of a number to a given base is that power to which the base must be raised to give the number'

    In other words, l = \log_a x \iff a^l = x

    So you'd do number 1, for instance, like this:

    l = \log_ax

    \Rightarrow a^l = x

    \Rightarrow (a^l)^k = x^k

    \Rightarrow a^{lk} = x^k

    \Rightarrow \log_ax^k = lk = k\log_ax

    Do you want to have a go at the others in the same way?

    Grandad
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