Calculus I · 2A · lesson

Derivatives of General Logarithms

Learn derivatives of general logarithms with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

Course progress26 of 49 core pages26 of 49

Section overview

Trigonometric, exponential, and logarithmic functions

What this section is building

Learn derivatives of general logarithms with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

Notice

Special-function rules preserve recognizable shapes while scaling them by a function-specific factor.

Decide

Identify the function family first, then check whether a composition requires the chain rule too.

Avoid

Using a power rule on an exponential or forgetting base and domain conditions for logarithms.

Concept

Learning objectives

Differentiate $\log_a x$ and use change of base.

Logarithms with Any Positive Base

Bridge

Base changes scale, not the basic reciprocal shape

At the same positive input, every logarithm has a derivative proportional to $1/x$ . The factor $1/\ln a$ stretches the slope vertically. Bases larger than one give positive slopes; bases between zero and one give negative slopes. This lets a graph check the sign of a symbolic answer before any algebra is simplified.

Explanation

Before the formulas

The formulas in Derivatives of General Logarithms are easiest to remember when connected to graph behavior. Trigonometric derivatives encode phase and sign patterns, exponential derivatives encode proportional growth, and logarithmic derivatives encode reciprocal sensitivity. The formulas are not unrelated entries in a table.

Keep domain and units visible. Trigonometric derivative formulas use radians. Logarithms require positive real inputs unless a different domain has been explicitly introduced. General exponential and logarithmic bases contribute constants such as $\ln a$ . These details are small on the page and decisive in a correct solution.

Explanation

Change the base before differentiating

Every logarithm can be written as a constant multiple of the natural logarithm. Since constants pass through differentiation, $\log_a x$ inherits the derivative of $\ln x$ with a scale factor $1/\ln a$ .

This is a useful example of mathematical economy: one well-understood function and one algebraic identity replace a separate derivative rule for every possible base.

Changing the logarithm base changes only a constant scale factor. Because

\log_a x=\frac{\ln x}{\ln a},

all logarithmic derivatives reduce to the natural-log rule. This is another example of calculus rewarding algebraic structure instead of brute-force memorization.

A derivative written in terms of $1/x$ should be interpreted carefully near zero. The rate becomes large because logarithmic outputs change rapidly when their positive input is small.

For $a>0$ , $a\ne1$ ,

\log_a x=\frac{\ln x}{\ln a}.

Since $\ln a$ is a constant,

\boxed{\frac{d}{dx}\log_a x=\frac1{x\ln a}}.

Guided walkthrough

Differentiate a common logarithm

Differentiate $f(x)=\log_{10}x$ .

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A base less than one

If $0<a<1$ , then $\ln a<0$ . Therefore $1/(x\ln a)<0$ , which correctly shows that $\log_a x$ is decreasing.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Logarithms with Any Positive Base

Bridge

Base changes scale, not the basic reciprocal shape

Explanation

Before the formulas

Explanation

Change the base before differentiating

This is a useful example of mathematical economy: one well-understood function and one algebraic identity replace a separate derivative rule for every possible base.

Changing the logarithm base changes only a constant scale factor. Because

\log_a x=\frac{\ln x}{\ln a},

all logarithmic derivatives reduce to the natural-log rule. This is another example of calculus rewarding algebraic structure instead of brute-force memorization.

A derivative written in terms of

1/x

should be interpreted carefully near zero. The rate becomes large because logarithmic outputs change rapidly when their positive input is small.

For

a>0

a\ne1

\log_a x=\frac{\ln x}{\ln a}.

Since

\ln a

is a constant,

\boxed{\frac{d}{dx}\log_a x=\frac1{x\ln a}}.

Guided walkthrough

Differentiate a common logarithm

Differentiate $f(x)=\log_{10}x$ .

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A base less than one

If $0<a<1$ , then $\ln a<0$ . Therefore $1/(x\ln a)<0$ , which correctly shows that $\log_a x$ is decreasing.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Logarithms with Any Positive Base

Related resources

Original instruction with traceable references.

Logarithms with Any Positive Base

Related resources

Original instruction with traceable references.