Derivative or Differentiation of Logarithmic Function as the name suggests, explores the derivatives of log functions with respect to some variable. As we know, derivatives are the backbone of Calculus and help us solve various real-life problems. Derivatives of the log functions are used to solve various differentiation of complex functions involving logarithms. The differentiation of logarithmic functions makes the product, division, and exponential complex functions easier to solve.

This article deals with all the information needed to understand the Derivative of the Logarithmic Function in plenty of detail including all the necessary formulas, and properties. We will also learn about the problem with their solutions as well as FAQs and practice problems on Differentiation of Log functions.

## Table of Content

- What are Logarithmic Functions?
- What is Derivative of Logarithmic Function?
- Derivative of Logarithmic Function Formula
- Proof of Derivative of Logarithmic Function Using First Principle
- Logarithmic Differentiation

## What are Logarithmic Functions?

The function that is the inverse of the exponential function is called the logarithmic function. It is represented as log_{b}x, where b is the base of the log. The value of x is the value which equals the base of the logarithm raised to a fixed number y, thus, the general form of a logarithmic function is:

y = log_{b}xWhere,

is the logarithm of x,yis the base of the logarithm, andbis the input of the logarithm.x

** Note:** As we know, logarithms and exponentials are related to each other such that If y = log

_{b}x then, x = b

^{y}.

** Learn more about **Logarithms

**.**### Properties of Logarithmic Function

Some properties of the logarithmic function are listed below:

- log (XY) = log X + log Y
- log (X / Y) = log X – log Y
- log X
^{Y}= Y log X - log
_{Y}X = ln X / ln Y

** Read more about **Log Rules

**.**## What is Derivative of Logarithmic Function?

The derivative of the logarithmic functions is solved using the properties of the logarithms and chain rule. It is used to solve complex functions which cannot be solved directly. By using the logarithmic functions, the complex functions can be simplified and can be evaluated easily.

Mostly, the exponential functions use the derivative of the logarithmic functions to get the solution of the complex functions. The functions of the form f(x)^{g(x)} can be easily evaluated using the derivative of the logarithms.

## Derivative of Logarithmic Function Formula

There are three formulas for the derivatives of the logarithmic functions. The following are the formulas for the derivative of the logarithmic functions.

- Derivative of ln x
- Derivative of log
_{a}x - Derivative of ln f(x)

Let’s discuss these formulas in detail.

### Derivative of ln x

The derivative of ln x evaluates to the reciprocal of x. The formula for the derivative of ln x is given below:

(d / dx) ln x = 1 / xWhere, x > 0

### Derivative of log_{a}x

The derivative of log with base a and value x evaluates to the reciprocal of the product of x and ln a. The formula for the derivative of log_{a}x is given below:

(d/dx) log_{a}x = 1 / [x ln a]Where, a ≠ 1

### Derivative of ln f(x)

The derivative of ln f(x) evaluates to the derivative of f(x) divided by f(x). The formula for the derivative of ln f(x) is given below:

(d/dx) ln f(x) = f'(x) / f(x)Where,

is any function of x, andf(x)is derivative of function of x.f'(x)

## Proof of Derivative of Logarithmic Function Using First Principle

Using the first principle of derivative

(d / dx) f(x) = lim

_{h→0}[{f(x +h) – f(x)} / {(x + h) – x}]Here, f(x) = ln x

(d / dx) f(x) = lim

_{h→0}[{ln(x +h) – ln(x)} / h]⇒ (d / dx) f(x) = lim

_{h→0}[{ln{(x +h)/(x)}} / h]⇒ (d / dx) f(x) = lim

_{h→0}[{ln(1 + (h / x))} / h]Now, putting (h / x) = (1 /n) and as limit h→0 then, (1 / n) → ∞

(d / dx) f(x) = lim

_{n→∞ }(n / x) [ln(1 + (1 / n))]⇒ (d / dx) f(x) = lim

_{n→∞}(1 / x) ln(1 + (1 / n))^{n}The value of lim

_{n→∞}ln(1 + (1 / n))^{n }= e(d / dx) f(x) = (1 / x) ln e . . . (1)

⇒ (d / dx) ln x = 1 / xFor (d / dx) log

_{a}x = 1 / (x ln a) putting in 1(d / dx) log

_{a}x = (1 / x) log_{a}e⇒ (d / dx) log

_{a}x = (1 / x) ln e /ln a [log_{a}e = ln e / ln a ]

⇒ (d / dx) log_{a}x = 1 / (x ln a)

## Logarithmic Differentiation

Logarithmic Differentiation uses the chain rule of differentiation with the differentiation formula of the log, and it helps us differentiate complex functions with ease. There are three forms of logarithmic differentiation i.e., differentiation of ln x, differentiation of log_{a}x and differentiation of ln f(x) whose differentiation formulas are mentioned above.

Let’s consider an example for Logarithmic Differentiation.

**Example: Find the derivative of log**_{3}**(x)**

**Solution:**

Let f(x) = log

_{3}(x)⇒ f'(x) = (d /dx) [log

_{3}x]⇒ f'(x) = 1 / [xln 3]

** Read more about, **Logarithmic Differentiation

**.****Also,Check**

- Inverse Trig Derivatives
- Differentiation and Integration Formula
- Derivative of Exponential Function

## Solved Examples on Derivatives of Logarithmic Function

**Example 1: Evaluate:**

**(i) Derivative of log 2**

**(ii) Derivative of log 3**

**(iii) Derivative of log 5**

**(iv) Derivative of log 10**

**Solution:**

As Derivative of log 2, Derivative of log 3, Derivative of log 5, and Derivative of log 10 are all constant values, and derivative of any constant is 0.

Thus, Derivative of log 2, Derivative of log 3, Derivative of log 5, and Derivative of log 10 are all equals to 0.

**Example 2: Find the following derivatives.**

**(i) Derivative of log 2x**

**(ii) Derivative of log**_{10}**x**

**(iii) Derivative of log y**

**Solution:**

(i) Derivative of log2x⇒ (d / dx) [log 2x] = (d / dx) [log 2x](d / dx) [2x]

⇒ (d / dx) [log 2x] = 2 / (2x)

⇒ (d / dx) [log 2x] = 1 / x

(ii) Derivative of log_{10}x⇒ (d / dx) [log

_{10}x] = 1 / [x ln 10]

(iii) Derivative of log y⇒ (d / dx) [log y] = [1 / y](dy / dx)

**Example 3: Find the derivative of ln (x**^{2}** + 4)**

**Solution:**

Let p(x) = ln(x

^{2}+ 4)⇒ p'(x) = (d / dx)[ln(x

^{2}+ 4)]By using formula

(d / dx) ln f(x) = f'(x) / f(x)

⇒ p'(x) = (2x) /(x

^{2}+ 4)

**Example 4: Evaluate: ln[(x**^{2}**sinx) / (2x + 1)]**

**Solution:**

Let f(x) = ln[(x

^{2}sinx) / (2x + 1)]By using the properties of ln

f(x) = 2lnx + ln(sinx) – ln(2x +1)

Now, differentiating,

f'(x) = 2(d /dx)lnx + (d / dx)[ln(sinx)] – (d / dx)[ln(2x +1)]

⇒ f'(x) = (2 / x) + (cos x / sin x) – [2 / (2x +1)]

⇒ f'(x) = (2 / x) + cot x – [2 / (2x +1)]

**Example 5: Find the slope of the line tangent of the graph of y = log**_{2}**(3x +1) at x = 1.**

**Solution:**

The slope of line tangent of a graph is given by (dy / dx)

(dy / dx) = (d / dx) [log

_{2}(3x +1)]⇒ (dy / dx) = 3 / [(3x +1)ln2]

At x = 1

⇒ (dy / dx)

_{at x =1}= 3 / [(3(1) +1)ln2]⇒ (dy / dx)

_{at x =1}= 3 / [4 ln2]⇒ (dy / dx)

_{at x =1}= 3 / [ln2^{4}]⇒ (dy / dx)

_{at x =1}= 3 / [ln16]

**Example 6: Evaluate the derivative: y = (2x**^{4 }**+ 1)**^{tanx}**.**

**Solution:**

Taking logarithm both sides

ln y = ln (2x

^{4}+ 1)^{tanx}⇒ ln y = tanx [ln (2x

^{4}+ 1)]Differentiating

(d / dx)ln y = (d / dx)[tanx [ln (2x

^{4}+ 1)]]Applying formula and product rule

(1 / y) (dy / dx) = [(d / dx)tanx {ln (2x

^{4}+ 1)}] + [tan x (d / dx){ln (2x^{4}+ 1)}]⇒ (1 / y) (dy / dx) = [sec

^{2}x ln (2x^{4}+ 1)] + [tan x {8x^{3}/ (2x^{4}+ 1)}]⇒ (dy / dx) = (2x

^{4}+ 1)^{tanx}[sec^{2}x ln (2x^{4}+ 1) + tan x {8x^{3}/ (2x^{4 }+ 1)}]

**Example 7: Find the derivative y = x**^{x}**.**

**Solution:**

y = x

^{x}Taking logarithm both sides

ln y = ln x

^{x}By logarithm properties

ln y = x lnx

Differentiating

(d /dx) ln y = (d /dx) [x ln x]

Applying product rule

(1 /y) (dy /dx) = (d /dx) (x) (ln x) + x(d /dx)( ln x)

⇒ (1 /y) (dy /dx) = ln x + x( 1/x)

⇒ (dy /dx) = y(ln x + 1)

⇒ (dy /dx) = x

^{x}(ln x + 1)

## Practice Problems on Differentiation of Logarithmic Function

** Problem 1: **Calculate the derivative of g(x) = 3ln(2x).

** Problem 2: **Determine the derivative of h(x) = ln(5x

^{2}).

** Problem 3: **Find the derivative of p(x) = ln(4x

^{3}+ 2x).

** Problem 4: **Calculate the derivative of q(x) = ln(1/x).

** Problem 5: **Determine the derivative of r(x) = ln(3x

^{2}– 7x + 1).

** Problem 6: **Find the derivative of s(x) = 2ln(sqrtx).

** Problem 7: **Calculate the derivative of t(x) = ln(x

^{3}+ 4x

^{2}– 2x + 1).

** Problem 8: **Determine the derivative of u(x) = ln(e

^{x}+ 1).

** Problem 9: **Find the derivative of v(x) = ln(2x

^{3}– 3x

^{2}+ 5x – 7).

## Derivatives of Logarithmic Function – FAQs

### 1. What is Log Function?

A logarithm (log) function is a mathematical operation that calculates the exponent to which a specified base must be raised to obtain a given number.

### 2. What is Derivative of Log Function?

The derivative of the natural logarithm ln x is given by the reciprocal of x. For the log with base a and function x given by the reciprocal of product of ln a and x.

### 3. What is Derivative of log x?

The derivative of log x is 1 /x.

### 4. What are the Formulas for Derivative of Log Function?

The formula for the derivative of logarithmic function

- (d / dx) ln x = 1 / x where, x >0
- (d / dx) log
_{a}x = 1 / (x ln a) where, a ≠ 1- (d / dx) ln f(x) = f'(x) / f(x)

### 5. How do you Find the Derivative of a Logarithm with a base Other than ‘e’?

To find the derivative of a logarithm with a base other than ‘e’ (natural logarithm), use the chain rule. For a logarithm with base ‘a’:

(d / dx) log_{a}x = 1 / (x ln a) where, a ≠ 1

### 6. How do you Differentiate Composition of a Logarithm, like ln(f(x))?

To differentiate a composition of logarithm, like ln(f(x)), you can use the chain rule. The derivative is:

(d / dx) ln f(x) = f'(x) / f(x)

### 7. What is the Second Derivative of ln(x)?

The second derivative of ln(x) is obtained by taking the derivative of its first derivative.

Since d/dx[ln(x)] = 1/x, the second derivative is:

d^{2}/dx^{2}[ln(x)] = d/dx[1/x] = -1/x^{2}

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Last Updated : 22 Sep, 2023

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I have extensive expertise in calculus and mathematical analysis, with a strong focus on derivatives and logarithmic functions. Throughout my academic and professional journey, I've delved into the intricate details of these mathematical concepts, applying them to solve real-life problems and complex functions.

Now, let's break down the key concepts covered in the article on the Derivative or Differentiation of Logarithmic Function:

### Logarithmic Functions:

Logarithmic functions are inverses of exponential functions, represented as log_b(x), where b is the base of the logarithm. The general form is y = log_bx, where y is the logarithm of x.

### Properties of Logarithmic Function:

- log (XY) = log X + log Y
- log (X / Y) = log X – log Y
- log XY = Y log X
- log YX = ln X / ln Y

### Derivative of Logarithmic Function:

The derivatives of logarithmic functions are crucial for solving complex functions involving logarithms. Three key formulas are provided:

- Derivative of ln x: (d / dx) ln x = 1 / x, where x > 0.
- Derivative of log_a x: (d / dx) log_a x = 1 / (x ln a), where a ≠ 1.
- Derivative of ln f(x): (d / dx) ln f(x) = f'(x) / f(x), where f(x) is any function of x.

### Proof of Derivative of Logarithmic Function Using First Principle:

The first principle of derivative is applied to prove the derivative of ln x, showcasing the limit as h approaches 0.

### Logarithmic Differentiation:

Logarithmic Differentiation utilizes the chain rule and differentiation formula of the log to simplify the differentiation of complex functions. Three forms are discussed: ln x, log_a x, and ln f(x).

### Solved Examples on Derivatives of Logarithmic Function:

Several examples demonstrate the application of derivative formulas to find the slope of tangent lines for given logarithmic functions.

### Practice Problems on Differentiation of Logarithmic Function:

A set of practice problems is provided, covering various scenarios of logarithmic functions for readers to apply their understanding and enhance their skills.

This comprehensive coverage ensures a deep understanding of the Derivative of Logarithmic Function, making it a valuable resource for students and enthusiasts alike. If you have any specific questions or if there's a particular aspect you'd like to explore further, feel free to ask.