Please see the attachment:
(a) 1. Set right hand side equal to x. We will show that x is the left hand side also.
2. Use the power property of logarithms.
3. Exponentiate both sides with base b.
4. Use a property of powers: a^(m+n)=a^m*a^n
5. Note that exponents and logarithms are inverse operations.
6. Take a logarithm of base b of both sides.
7. Exponents and logarithms are inverses.
8. Substitute for x from (1).
(b) If you really wanted the product rule log(mn)=log(m)+log(n) you would use a similar procedure.
log(mn)=x=log(m)+log(n) as required.