Derivatives of Logs

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Derivatives of Logs

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Formulas and Examples

We defined log functions as inverses of exponentials: \begin{eqnarray*} y = \ln(x) &\Longleftrightarrow & x = e^y \cr y = \log_a(x) & \Longleftrightarrow & x = a^y. \end{eqnarray*} Since we know how to differentiate exponentials, we can use implicit differentiation to find the derivatives of $\ln(x)$ and $\log_a(x)$. The videos below walk us through this process.

The end results are:

$$\frac{d}{dx} \ln(x) = \frac{1}{x}, \qquad \frac{d}{dx}\log_a(x) = \frac{1}{x \ln(a)}.$$

The derivative of $\ln(x)$:

\begin{eqnarray*} y & = & \ln(x) \cr x & = & e^y \cr 1 & = & \frac{d }{dx}\left(e^y\right) \cr 1 & = & e^y \frac{dy}{dx} \cr \frac{dy}{dx} & = & \frac{1}{e^y} \cr \frac{dy}{dx} & = & \frac{1}{x}
\end{eqnarray*}since $e^y=x$ (see above).

The derivative of $\log_a(x)$:

\begin{eqnarray*} y & = & \log_a(x) \cr x & = & a^y \cr 1 & = & \frac{d}{dx} \left( a^y\right)\cr 1 & = & a^y \ln(a) \frac{dy}{dx} \cr \frac{dy}{dx} & = & \frac{1}{a^y \ln(a)} \cr \frac{dy}{dx} & = & \frac{1}{x \ln(a)}. \end{eqnarray*} since $a^y=x$ (see above).