Derivatives of Logarithmic Functions

Properties of Logarithms

Before we look at derivatives of logarithmic functions, it is important to review some very useful properties of logarithms.

Property 1 (Sum Law): Provided that $B, C > 0$, $\log_a B + \log_a C = \log_a (BC)$.
Property 2 (Difference Law): Provided that $B, C > 0$, $\log_a B - \log_a C = \log_a \left ( \frac{B}{C} \right)$.
Property 3 (Power Law): Provided that $B > 0$ and $k \in \mathbb{R}$, $\log_{a}(B^k) = k \log_a B$.
Property 4 (Change of Base Law): Provided that $a, C > 0$, $\log_{a} B = \frac{\log_{c} B}{\log_{c} a}$.

Furthermore, it is also important to define the natural logarithm:

Definition: The Natural Logarithm denoted $\ln x = \log_e x$ is the logarithm of base $e \approx 2.71828...$.

Domain and Range of Logarithmic Functions

Recall that the domain of a logarithmic function is limited. Let's look at some function $f(x) = \log x \Leftrightarrow 10^y = x$. We can clearly see that the function $g(x) = 10^x$ is the inverse of $f$. Recall that if $f$ and $g$ are inverses of eachother, then $D(f) = R(g)$ and $D(g) = R(f)$. From this, we note that $D(g) = (-\infty, \infty) = R(f)$, and similarly, $R(g) = (0, \infty) = D(f)$. Therefore, the domain of a logarithmic function $f$ is $(0, \infty$, and the range is $(-\infty, \infty)$.

We will now proceed in learning how to evaluate the derivative of a logarithmic function.

Derivatives of Logarithmic Functions.

Theorem 1: If $f$ is a logarithmic function such that $f(x) = \log_a x$, then the derivative $f'(x) = \frac{1}{x \ln a}$.

Example 1

Determine the derivative of the function $f(x) = \log_{5} x$.

Applying the formula we get that:

\begin{align} \frac{d}{dx} \log_{5}(x) = \frac{1}{x \ln(5)} \end{align}

Natural Logarithm Derivative

Theorem 2: If $f(x) = \ln x = \log_{e} x$, then $f'(x) = \frac{1}{x}$.

If we apply the natural logarithm to the formula we mentioned earlier and note that $\ln(e) = 1$, then we get that:

\begin{align} \frac{d}{dx} \ln x = \frac{1}{x \ln(e)} \\ \frac{d}{dx} \ln x = \frac{1}{x} \end{align}
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