Calculus
Textbooks
Boundless Calculus
Inverse Functions and Advanced Integration
Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
Calculus Textbooks Boundless Calculus Inverse Functions and Advanced Integration Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
Calculus Textbooks Boundless Calculus Inverse Functions and Advanced Integration
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 9
Created by Boundless

Derivatives of Logarithmic Functions

The general form of the derivative of a logarithmic function is $\frac{d}{dx}\log_{b}(x) = \frac{1}{xln(b)}$.

Learning Objective

  • Solve for the derivative of a logarithmic function


Key Points

    • The derivative of natural logarithmic function is $\frac{d}{dx}\ln(x) = \frac{1}{x}$.
    • The general form of the derivative of a logarithmic function can be derived from the derivative of a natural logarithmic function.
    • Properties of the logarithm can be used to to differentiate more difficult functions, such as products with many terms, quotients of composed functions, or functions with variable or function exponents.

Terms

  • logarithm

    the exponent by which another fixed value, the base, must be raised to produce that number

  • e

    the base of the natural logarithm, $2.718281828459045\dots$


Full Text

Here, we will cover derivatives of logarithmic functions. First, we will derive the equation for a specific case (the natural log, where the base is $e$), and then we will work to generalize it for any logarithm.

Let us create a variable $y$ such that $y = \ln (x)$.

It should be noted that what we want is the derivative of y, or $\frac{dy}{dx}$.

Next, we will raise both sides to the power of $e$ in an attempt to remove the logarithm from the right hand side:

$e^{y} = x$

Applying the chain rule and the property of exponents we derived earlier, we can take the derivative of both sides:

$\dfrac{dy}{dx} \cdot e^{y} = 1$

This leaves us with the derivative

$\dfrac{dy}{dx} = \dfrac{1}{e^{y}}$

Substituting back our original equation of $x = e^{y}$, we find that 

$\dfrac{d}{dx}\ln(x) = \dfrac{1}{x}$

If we wanted, we could go through that same process again for a generalized base, but it is easier just to use properties of logs and realize that 

$\log_{b}(x) = \dfrac{\ln(x)}{\ln(b)}$

Since $\frac{1}{\ln(b)}$ is a constant, we can take it out of the derivative:

$\dfrac{d}{dx}\log_{b}(x) = \dfrac{1}{\ln(b)} \cdot \dfrac{d}{dx}\ln(x)$,

which leaves us with the generalized form of:

$\dfrac{d}{dx}\log_{b}(x) = \dfrac{1}{x \ln(b)}$

We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such as products with many terms, quotients of composed functions, or functions with variable or function exponents. We do this by taking the natural logarithm of both sides and re-arranging terms using the following logarithm laws:

  • $\log \left(\dfrac{a}{b}\right) = \log (a) - \log (b)$
  • $\log(a^{n}) = n \log(a)$
  • $\log(a) + \log (b) = \log(ab)$

and then differentiating both sides implicitly, before multiplying through by $y$.

[ edit ]
Edit this content
Prev Concept
Logarithmic Functions
The Natural Logarithmic Function: Differentiation and Integration
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.