Problem: Logarithm of a product

Problem

Prove that \(log \left( xy \right)=log \left( x \right)+log \left( y \right)\) .

Hint: Start with the identity \(xy=xy\) (restricted to \(x>0,y>0\) ).

• Explain why on the left-hand side, \(xy=e^{log \left( xy \right)}\) .
• On the right-hand side, explain why \(xy=e^{log \left( x \right)}e^{log \left( y \right)}\)
• Explain why on the right-hand side, \(xy=e^{log \left( x \right)+log \left( y \right)}\)
• Combining a) and c), \(e^{log \left( xy \right)}=e^{log \left( x \right)+log \left( y \right)}\) which implies that \(log \left( xy \right)=log \left( x \right)+log \left( y \right)\) ( \(e^x\) is strictly increasing)