Use the product rule: (log_b(MN)=log_b M+log_b N)
Use the quotient rule: (log_bleft(frac{M}{N}right)=log_b M-log_b N)
Use the power rule: (log_b(M^k)=klog_b M)
Rewrite roots as fractional exponents before expanding: (log_b(sqrt[n]{M})=log_b(M^{1/n})=frac{1}{n}log_b M)
Apply the rules repeatedly to fully expand complex expressions
Keep the logarithm base the same throughout
Expand only when the arguments are multiplied, divided, or raised to powers
Do not split sums or differences inside a logarithm: (log_b(M+N)) and (log_b(M-N)) cannot be expanded using log rules
Example: (log_bleft(frac{x^2y}{z^3}right)=2log_b x+log_b y-3log_b z)
Example: (log_bleft(sqrt{frac{a^3}{bc}}right)=frac{3}{2}log_b a-frac{1}{2}log_b b-frac{1}{2}log_b c)