Problem: Logarithm of a power

Problem

Prove that \(log \left( x^y \right)=ylog \left( x \right)\)

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

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