Rules of logarithms

To manipulate logarithmic expressions and solve equations, we use several key rules.

Formula: Logarithmic rules

For positive x,y,a and a1:

  1. Product Rule: loga(xy)=logax+logay
  2. Quotient Rule: loga(xy)=logaxlogay
  3. Power Rule: loga(xn)=nlogax

Special values:

  • logaa=1
  • loga1=0
  • In particular, lne=1 and ln1=0.

Example A: Simplifying log expressions

Question

Simplify the expression: log23+2log210log2150.


Solution

log23+2log210log2150=log23+log2(102)log2150(Power rule)=log23+log2100log2150=log2(3×100)log2150(Product rule)=log2300log2150=log2(300150)(Quotient rule)=log22=1