Derivatives of Logarithmic Functions (ln and log)

Logarithms appear in many contexts, from growth/decay models to information theory. Their derivatives are straightforward if you remember two core formulas: the natural logarithm $\ln(u)$ and the logarithm to a general base $\log_a(u)$.

📜 Core Formulas

Natural Logarithm

For $u = u(x) > 0$:

$$ \frac{d}{dx}\big(\ln(u)\big) = \frac{u'(x)}{u(x)} $$

Logarithm to a General Base

For $u = u(x) > 0$ and base $a > 0,\ a \neq 1$:

$$ \frac{d}{dx}\big(\log_a(u)\big) = \frac{u'(x)}{u(x)\,\ln(a)} $$

💡 Change of base: $\log_a(u) = \dfrac{\ln(u)}{\ln(a)}$. This identity explains the extra factor $\ln(a)$ in the denominator. Domain: The argument must be positive ($u(x) > 0$).

🪜 Application in 3 Simple Steps

Use the chain rule mindset: treat the log as the outer function and your argument as the inner function.

Identify the argument: Write your function as $\ln(u(x))$ or $\log_a(u(x))$ with $u(x) > 0$.
Differentiate the outer function: For $\ln(u)$, the outer derivative is $\dfrac{1}{u}$; for $\log_a(u)$, it is $\dfrac{1}{u\ln(a)}$.
Multiply by the inner derivative: Multiply by $u'(x)$ and simplify if helpful.

✏️ Examples for Illustration

Example 1: $\boldsymbol{\ln(x)}$

Given the function: $f(x) = \ln(x)$ (for $x > 0$)

Derivative: $$ f'(x) = \frac{1}{x}. $$

Example 2: $\boldsymbol{\ln(3x^2 + 1)}$

Given the function: $f(x) = \ln(3x^2 + 1)$ (domain: $3x^2 + 1 > 0$ — always true)

$u(x) = 3x^2 + 1 \Rightarrow u'(x) = 6x$

Derivative: $$ f'(x) = \frac{u'(x)}{u(x)} = \frac{6x}{3x^2 + 1}. $$

Example 3: $\boldsymbol{\log_5(x)}$

Given the function: $f(x) = \log_{5}(x)$ (for $x > 0$)

Derivative: $$ f'(x) = \frac{1}{x \,\ln(5)}. $$

Example 4: $\boldsymbol{\log_a(x^3)}$

Given the function: $f(x) = \log_{a}(x^3)$ with $a > 0,\ a \neq 1$ and $x \neq 0$

$u(x) = x^3 \Rightarrow u'(x) = 3x^2$

Derivative: $$ f'(x) = \frac{3x^2}{x^3\,\ln(a)} = \frac{3}{x\,\ln(a)}. $$

Example 5: $\boldsymbol{\ln(\sin x)}$ (chain rule)

Given the function: $f(x) = \ln(\sin x)$ with $\sin x > 0$ (domain restriction)

$u(x) = \sin x \Rightarrow u'(x) = \cos x$

Derivative: $$ f'(x) = \frac{\cos x}{\sin x} = \cot x. $$

Example 6: $\boldsymbol{\log_{10}(2x+5)}$

Given the function: $f(x) = \log_{10}(2x+5)$ with $2x+5 > 0$

$u(x) = 2x+5 \Rightarrow u'(x) = 2$

Derivative: $$ f'(x) = \frac{2}{(2x+5)\,\ln(10)}. $$