First of all, recall that the definition of a logarithm allows us to rewrite an exponential equation using logarithms and vice versa:

\[ \solve{} x=B^y\iff y=\log_B(x) \]

Once we have this fact in hand, the Laws of Logarithms some end up being the exponential rules simply converted into logarithmic form while others have some very clear analagous forms. So, let's review all of the exponential rules first:

\[ \solve{ B^1 &=&B\\\\ B^0 &=&1\\\\ B^m\times B^n&=&B^{m+n}\\\\ \dfrac{B^m}{B^n}&=&B^{m-n}\\\\ \left(B^m\right)^m &=&B^{{mn}}\\\\ B^{-m} &=&\dfrac{{1}}{B^m}\\\\ \sqrt[n]{{B}}&=&B^{\frac{{1}}{{n}} } } \]

We can rewrite the first two directly into logarithmic form:

\[ \solve{ 1 &=& \log_B(B)\\\\ 0 &=& \log_B(1)} \]

These two are very important properties of logarithms which you will need to know in order to simplify expressions containing logarithms. The remaining properties can be shown to be derivatives of either earlier properties or the original exponential rules.

\[ \solve{ \text{Power Rule}&:&\log_B(A^n)=n\log_B(A)\\\\ \text{Product Rule}&:&\log_B(A\times C) = \log_B(A)+\log_B(C)\\\\ \text{Quotient Rule}&:&\log_B\left(\frac{{A}}{{C}}\right)=\log_B(A)-\log_B(C) } \]

The actual proofs rely on techniques like induction, so I do not plan to go into full detail here, but I would encourage you to think about why these rules are valid.