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,

$\displaystyle \log_a x^k = k \log_a x $

2] Given that x > 0, y > 0, show that

$\displaystyle \log_a \frac{x}{y} = \log_a x - \log_a y $

3] Given that x > 0, y > 0, show that

$\displaystyle \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?