Derivative of Log x: Formula and Proof

Derivative of log x is 1/x. Log x Derivative refers to the process of finding change in log x function to the independent variable. The specific process of finding the derivative for log x functions is referred to as logarithmic differentiation.

Let’s know more about Derivative of Log x formula and proof in detail below.

Derivative of log x

Derivative of log x is 1/x. It represents the rate of change of log x with respect to the change in the value of independent variable x. This means that the slope of the tangent line to the graph of log x at any point is 1/x in the base of 10..

Note: If we change the base from e to 10, derivative of log changes to 1/x as ln e = 1.

Derivative of log x Formula

The formula for the derivative of log x is given by:

(d/dx)[log x] = 1 / (x ln 10)

OR

( log x)’ = 1/ (x)

Proof of Derivative of log x

The derivative of log x can be proved using the following 3 ways:

• By First Principle of Derivative
• By Implicit Differentiation Method
• By ln x Formula

Derivative of log x by First Principle of Derivative

To prove derivative of log x using First Principle of Derivative, we will use basic limits and log formulas which are listed below:

• logₐ m – logₐ n = logₐ (m/n)
• m logₐ a = logₐ a^m
• a^mn = (a^m)ⁿ
• logₐ a^m = m logₐ a
• lim_t→0 [(1 + t)^1/t] = e

Let’s start the proof for the derivative of log x , assume f(x)=log_ax

By First Principle of Derivative

f'(x) = lim_h→0 [f(x + h) – f(x)] / h

Since f(x) = logₐ x, we have f(x + h) = logₐ (x + h).

Substituting these values in the equation of first principle,

f'(x) = lim_h→0 [logₐ (x + h) – logₐ x] / h

⇒ f'(x) = lim_h→0 [logₐ [(x + h) / x] ] / h { By 1 }

⇒ lim_h→0 [logₐ (1 + (h/x))] / h

Assume that h/x = t. From this, h = xt.

When h →0, h/x → 0 ⇒ t → 0.

Then the above limit becomes

f'(x) = lim_t→0 [logₐ (1 + t)] / (xt)

⇒ lim_t→0 1/(xt) logₐ (1 + t)

⇒ f'(x) = lim_t→0 logₐ (1 + t)^1/(xt) { By 2 }

⇒ f'(x) = lim_t→0 logₐ [(1 + t)^1/t]^1/x { By 3 }

⇒ f'(x) = lim_t→0 (1/x) logₐ [(1 + t)^1/t] { By 4 }

Here, the variable of the limit is ‘t’. So we can write (1/x) outside of the limit.

f'(x) = (1/x) lim_t→0 logₐ [(1 + t)^1/t] = (1/x) logₐ lim_t→0 [(1 + t)^1/t]

Using one of the formulas of limits,. Therefore,

f'(x) = (1/x) logₐ e { By 5 }

⇒ (1/x) (1/logₑ a) (because ‘a’ and ‘e’ are interchanged)

⇒ (1/x) (1/ ln a) (because logₑ = ln)

⇒ 1 / (x ln a)

Therefore for f(x)=log x , f'(x) =1/(x ln 10)

Derivative of log x by Implicit Differentiation Method

The derivative of log x can be proved using the implicit differentiation method. In this method, if we are given an implicit function, then we take the derivative on both sides of the equation with respect to the independent variable. We will use basic logarithmic formulas which are listed below:

• dy/dx(a^y) =a^y ln a
• y = log_ax ⇒ a^y=x

Let’s start the proof for the derivative of log x , assume f(x)=y=log_ax

By Implicit Differentiation Method

f(x)=y=log_ax

⇒ a^y=x

Taking derivative on both sides with respect to “x”

⇒ d/dx (a^y) = d/dx (x)

By using the chain rule,

(a^y ln a) dy/dx = 1 { By 1 }

⇒ dy/dx = 1/(a^y ln a)

But we have a^y = x. Therefore,

dy/dx = 1 / (x ln a)

Therefore for f(x)=log x , f'(x) = 1 / (x ln 10)

Derivative of log x by ln x Formula

We know that the derivative of ln x is 1/x. We can convert log into ln using change of base rule. We will be using basic log formulas which are listed below:

• log _a x = (logₑ x) / (logₑ a)
• logₑ = ln

Let’s start the proof for the derivative of log x , assume f(x)=log_ax

By change of base rule

f(x) = (logₑ x) / (logₑ a) { By 1 }

⇒ f(x) = (ln x) / (ln a) { By 2 }

Now we will find its derivative.

f'(x) = d/dx [(ln x) / (ln a)]

⇒ 1/ (ln a) d/dx (ln x)

⇒ 1 / (ln a) · (1/x)

⇒ 1 / (x ln a)

Therefore for f(x)= log x , f'(x) =1 / (x ln 10)

Facts about Logarithm

Some common facts about logarithm are:

• Logarithm generally can be of three types,
  • Common Logarithm: Base 10, denoted as log x or log₁₀ x
  • Natural Logarithm: Base e, denoted as ln x or log_e x
  • Binary Logarithm: Base 2, denoted as log₂ x
• log 1 in any base is always 0 i.e., log_a 1 = 0
• Logarithm of a with same base as a is always 1 i.e., log_a a = 1.

Learn More,

• Logarithmic Function
• Differentiation Formulas
• Logarithmic Differentiation

Solved Examples on Derivative of log x

Example 1: Find the derivative of 2log x.

Solution:

d/dx[ 2 log x ] = 2 /(x ln 10)

Example 2: Find the derivative of 10log x at x = 10.

Solution:

d/dx[ 10 log x ] = 10 /(x ln 10)

at x=10 , d/dx[ 10 log x ] = 10/(10 ln 10) = 1/ (ln 10)

Example 3: Find the derivative of x²log₂x .

Solution:

d/dx[ x²log₂x ] = 2x(log₂x) + x²(1/x ln 2)

formula,

d/dx[ log x² ] = ( 2x ) x ( 1/x²ln10 )

Practice Questions on Derivative of log x

Q1: Find the derivative of log x.

Q2: Find the derivative of x² log x².

Q3: Evaluate derivative of log sin x.

Q4: Evaluate the derivative of x log x. 

Q5: Find the derivative of x³ log cos x.

Derivative of log x – FAQs

What is Derivative?

The derivative of the function is defined as the rate of change of the function with respect to a variable.

What is Formula for Derivative of log x?

Formula for derivative of log x is (d/dx) ( log x) = 1 / ( x ln 10 )

What is Derivative of – log x?

The derivative of – log x is (d/dx) ( -log x) = -1 / ( x ln 10 )

What are Different Methods to Prove Derivative of log x ?

The different methods to prove derivative of log x are:

• By using the First Principle of Derivative
• By using Implicit Differentiation Method
• By using ln x Formula

What is Derivative of ln x?

The derivative of ln x is (d/dx) ( ln x) = 1 / x