Deriving the Derivative of \(\log_a x\)

Derivation of \(\dfrac{\mathrm d}{\mathrm dx}(\log_a x)=\dfrac1{x\ln a}\)

Use the change-of-base formula \(\log_a x=\frac{\ln x}{\ln a}\), where \(a>0\), \(a\ne1\), and \(a\) is constant.

Therefore \(\frac{\mathrm d}{\mathrm dx}(\log_a x)=\frac{\mathrm d}{\mathrm dx}\left(\frac{\ln x}{\ln a}\right)=\frac1{\ln a}\frac{\mathrm d}{\mathrm dx}(\ln x)\).

Since \(\frac{\mathrm d}{\mathrm dx}(\ln x)=\frac1x\), it follows that \(\frac{\mathrm d}{\mathrm dx}(\log_a x)=\frac1{x\ln a}\).