Derivatives Of Logarithmic Functions

Table of Contents

 

• Brief Review of Exponential Functions and Logarithmic Functions
• Derivatives of Logarithmic Functions - Formulas
• Examples

 

Brief Review of Exponential Functions and Logarithmic Functions

 

An exponential function is a function of the form:

 

\[\displaystyle a^{x} \]

 

where \(a\) is a non-zero number and \(x\) is a variable.

The logarithmic function is the inverse function of the exponential function. The logarithmic function is of the form:

 

\[\displaystyle \log_a (x)\]

 

where the base \(a\) is a non-zero number and \(x\) is a variable.

If we have the exponential function of \(\text{e}^{x}\) with Euler’s constant of \(e \approx 2.71828\) then the corresponding inverse would be \(\log_e(x) = \ln(x)\).

 

Derivatives of Logarithmic Functions - Formulas

 

Given that \(f(x)\) is of the form \(f(x) = \log_a(g(x))\) then the derivative \(f'(x)\) is:

 

\[\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{g(x)} \cdot \dfrac{d}{dx} g(x) \\ & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{g(x)} \cdot g'(x) \\ \end{array} \\ \]

 

A more simpler case is when we have \(f(x) = \log_a (x)\). The derivative would be:

 

\[\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{x} \cdot \dfrac{d}{dx} x \\ & = & \dfrac{1}{\text{ln} (a)} \cdot \dfrac{1}{x} \cdot 1 \\ & = & \dfrac{1}{\ln (a)} \cdot \dfrac{1}{g(x)} \\ \end{array} \\ \]

 

The most simplest and most common case is taking the derivative of \(f(x) = \ln(x)\).

 

\[\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{d}{dx} \ln(x) \\ & = & \dfrac{1}{\ln(e)} \cdot \dfrac{1}{x} \cdot \dfrac{d}{dx} x\\ & = & \dfrac{1}{1} \cdot \dfrac{1}{x} \cdot 1\\ & = & \dfrac{1}{x} \\ \end{array} \\ \]  

(Note that ln(e) = 1.)

 

Examples

 

Here are some examples of differentiating logarithmic functions.

 

Example One

 

Given that \(f(x) = \ln(2x)\) then the derivative is:

 

\[\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{d}{dx} \ln(2x) \\ & = & \dfrac{1}{\ln(e)} \cdot \dfrac{1}{2x} \cdot \dfrac{d}{dx} 2x\\ & = & \dfrac{1}{1} \cdot \dfrac{1}{2x} \cdot 2\\ & = & \dfrac{1}{x} \\ \end{array} \\ \]

 

Example Two

 

If we have \(f(x) = \log_5(x)\) then the derivative is:

 

\[\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\text{ln}(5) \cdot x} \cdot 1 \\ & = & \dfrac{1}{\ln(5) \cdot x} \\ \end{array} \\ \]  

Example Three

 

The derivative of the function \(f(x) = \log_{10}(x^2)\) is:

 

\[\displaystyle \begin{array} {lcl} f'(x) & = & \dfrac{1}{\ln(10) \cdot x^2} \cdot 2x \\ & = & \dfrac{2}{\ln(10) \cdot x} \\ \end{array} \\ \]

 

Example Four

 

This last example is more involved. Suppose that the function \(g(x)\) is \(g(x) = \text{ln}(\text{ln}(x^5))\). At first, you may be terrified. However, if you look at this more closely this is a case of using chain rule. You would need to differentiate multiple times. The derivative of \(g\) is as follows.

 

\[\displaystyle \begin{array} {lcl} g'(x) & = & \dfrac{d}{dx} \text{ln}(\text{ln}(x^5)) \\ & = & \dfrac{1}{\text{ln}(x^5)} \cdot \dfrac{d}{dx} \text{ln}(x^5) \\ & = & \dfrac{1}{\text{ln}(x^5)} \cdot \dfrac{1}{x^5} \cdot \dfrac{d}{dx} (x^5) \\ & = & \dfrac{1}{\text{ln}(x^5)} \cdot \dfrac{1}{x^5} \cdot 5x^4 \\ & = & \dfrac{5}{\text{ln}(x^5) \cdot x}\\ \end{array} \\ \]

 

When dealing with chain rule cases, go from the outside to the inside. From the first line to the second line the derivative of \(\ln( \cdot )\) is \(\dfrac{1}{\ln( \cdot )}\). Then we take the derivative of the inside which is \(( \cdot )\).