Derivatives of Logarithmic Functions

The central idea

For $x>0$, $d(\ln x)/dx=1/x$; more generally, where $u(x)\ne0$, $d(\ln|u|)/dx=u'/u$. For $a>0$, $a\ne1$, $d(\log_a x)/dx=1/(x\ln a)$. Logarithmic differentiation turns products, quotients, and variable powers into sums that are easier to differentiate.

Definitions, hypotheses, and notation

The formula $(\ln|u|)'=u'/u$ is local to intervals where $u$ never vanishes. It does not make $\ln|u|$ defined at a zero of $u$, and the derivative typically has a vertical blow-up there. For a logarithm without absolute value, one must also require $u>0$. Stating these domains prevents a formally correct derivative from being attached to nonexistent inputs.

Logarithmic differentiation is most efficient when a positive function contains many factors or a variable exponent. Log laws turn multiplication into addition and exponents into coefficients before differentiation. It is unnecessary for a simple $\ln(g(x))$, where direct use of $g'/g$ is shorter. Method choice depends on structure, not merely on seeing a logarithm.

Conceptual meaning

The derivative $1/x$ shows that logarithmic growth continually slows. The absolute value in $\ln|u|$ allows the derivative formula on intervals where $u$ is negative as well as positive, but never where $u=0$.

A dependable method and decision rule

1. Check the logarithm's domain before differentiating.
2. Identify the full argument $u(x)$.
3. Differentiate as $u'/u$, retaining any base factor $1/\ln a$.
4. For a complicated positive expression $y$, take $\ln$ of both sides and expand log laws.
5. After implicit differentiation, solve for $y'$ and substitute back for $y$.

Fully worked example

Graphical or geometric meaning

The graph of $\ln x$ is increasing and concave down on $x>0$. Its tangent slope $1/x$ is large near zero and approaches zero as $x$ grows. The vertical asymptote records the domain boundary.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate an expanded logarithmic expression and compare with the quotient $u'/u$.
• Confirm the derivative is defined only on intervals allowed by the original function.
• Use a sign check: $\ln x$ must have positive slope for $x>0$.

Domain and absolute value are part of the rule

For differentiable $u$ with $u\ne0$, $\frac d{dx}\ln|u|=u'/u$ on each interval where $u$ does not vanish. Writing $\ln u$ is valid only where $u>0$. Crossing a zero can split the formula into separate intervals even when the derivative expression looks simple. For base $a$, include $1/\ln a$. Logarithm laws may turn products into sums and powers into coefficients, but their domain conditions remain. Scaling gives a useful check: multiplying $u$ by a positive constant changes $\ln u$ by a constant, so it should not alter the derivative $u'/u$. Always report restrictions inherited from the original logarithm.

Practice

1. Differentiate $\ln(3x-1)$.
2. Differentiate $\log_2x$.
3. Differentiate $\ln|x^2-4|$ where defined.

Connections and next steps

• Derivatives of Exponential Functions
• Chain Rule
• Implicit Differentiation