Logarithmic Differentiation

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Mathematics is the abstract study of different topics such as quantity, number theory, structure (algebra). It is evolved from counting, measuring, and describing the shape of objects. The word Mathematics originates from the Greek word Mathema.

There are many branches of Mathematics such as algebra, geometry, trigonometry, calculus, probability, and statistics.

Logarithmic differentiation is a part of calculus. The technique is used in cases where it is easier to differentiate the logarithm of a function rather than the function itself. In some cases, it is easier to differentiate the logarithm of a given function than to differentiate it from the function itself.

It is a famous concept, and it applies to the majority of the non-zero functions. The only condition is that the non-zero functions should be differentiable in the future. Differentiation and integration are the two main concepts of calculus. Differentiation is used to study the small change of a quantity. Integration is different from differentiation. It is used to add small and discrete data and cannot be added singularly.

Derivative of Logarithm

When a logarithmic function is represented as:

  • f (x) = logb(x)

The derivative of a logarithmic function is given by:

  • f ' (x) = 1 / ( x ln(b) )

  1. Here, x is called as the function argument.

  2. b is the logarithm base.

  3. ln b is the natural logarithm of b.

We can differentiate log in this way.

The derivative of ln(x) is 1/x

This is the way of differentiating ln. The derivative of ln(x) is a well-known derivative.

Following are some of the examples of logarithmic derivatives:

1. Find the Value of dy/dx if,y=ex4

Solution: Given the function y=ex4

Taking the natural logarithm of both the sides we get,

ln y = ln ex4

ln y = x 4 ln e

ln y = x4

Now, differentiating both the sides w.r.t we get,

1ydydx = 4x3

⇒dy dx =y.4x3

⇒dydx =ex4×4x3

2. Find the Value of dy dx if y = 2x{cos x}.

Solution: Given the function y = 2x{cos x}

Taking logarithm of both the sides, we get

log y = log(2x{cos x})

⇒log y=log2+log x cosx(As log(mn)=logm+logn)

⇒logy=log2+cosx×logx(As logmn=nlogm)

Now, differentiating both the sides w.r.t by using the chain rule we get,


This is a way used for differentiating logarithmic functions.

y ``(x)=(logax)′=1 xlna. 

If a=e, we obtain the natural logarithm the derivative of which is expressed by the formula (lnx)′=1x, 

where the number M is equal to M=log10e≈0.43429.

We derived the formula (logax)′=1xlna from first principles using the derivative's limit definition.

This is the way of deriving logarithms.

Deriving log functions becomes possible because of the use of exponents.

Following are some of the log derivative rules:

Common Functions    

Function Derivative

  • Constant   c       0

  • Line  x       1

         ax     a

  • Square      x2     2x

  • Square Root       √x     (½)x-½

  • Exponential        ex     ex

         ax     ln(a) ax

  • Logarithms ln(x) 1/x

         loga(x)       1 / (x ln(a))

  • Trigonometry (x is in radians)       sin(x) cos(x)

         cos(x)        −sin(x)

         tan(x)        sec2(x)

  • Inverse Trigonometry sin-1(x)      1/√(1−x2)

         cos-1(x)     −1/√(1−x2)

         tan-1(x)     1/(1+x2)

Rules Function

  • Derivative

  • Multiplication by constant  cf      cf’

  • Power Rule         xn     nxn−1

  • Sum Rule  f + g f’ + g’

  • Difference Rule  f - g  f’ − g’

  • Product Rule      fg     f g’ + f’ g

  • Quotient Rule    f/g    (f’ g − g’ f )/g2

  • Reciprocal Rule  1/f    −f’/f2                 

  • Chain Rule

(as "Composition of Functions")   f º g  (f’ º g) × g’

Chain Rule (using ’ )    f(g(x))        f’(g(x))g’(x)

Chain Rule (using  ddx )        dy dx =  dy du du dx

  1. These are logarithmic differentiation rules.

The logarithmic function with base a (a>0, a≠1) and exponential function with the same base form a pair of mutually inverse functions; the log function's derivative is also found using the inverse function theorem. (logax)′=f′(x)=1φ′(y)=1(ay)′=1aylna=1alogaxlna=1xlna.

The differentiation of natural log ln(x) is 1 divided by x.

  1. Logarithmic differentiation steps are as follows:-

A natural log is supposed to be taken on both sides.

Use the property of the log of the product.

Differentiate on both sides. For every term on the right side of the equation, a chain rule should be used.

The last step is to multiply both sides by f(x).

Following are the logarithm derivative rules we always need to follow:-

The slope of a constant value (for example 3) is always 0.

The slope of a line like 2x is 2, or 3x is 3, etc. 

One can use logarithmic differentiation when applied to functions raised to the power of variables or functions. 

Logarithmic differentiation relies on the chain rule as well as the properties of the logarithm.

Differentiation in y

The derivative of function y = f(x) of a variable x is the measure of the rate at which the value y of the function changes concerning the change of the variable x. The derivative of ln y is 1/ (derivative of f = e\[^x\]) = 1/e\[^x\].

FAQ (Frequently Asked Questions)

1. What are Logarithms?

A logarithm is a mathematical function used to determine the number of times the base is multiplied by itself to reach another number. They relate geometric progressions to arithmetic progressions. Some common examples in day-to-day life are the spacing of guitar chords, earthquakes, windstorms, etc.

2. What is the Formula for Differential Log?

The differential log formula is as follows: - 

For a function


On taking absolute values, it becomes: -

ln |y|= ln |f(x)|

3. What is an Exponential Function?

The exponential function is incredibly unique in Mathematics. To form an exponential function, the independent variable should be the exponent. One of the exponential function parameters is that the action of an exponential function f(x) can be captured by the function machine metaphor, which takes inputs x and transforms them into the outputs f(x).

4. Why is it Necessary to Take the Derivative of a Function?

The derivative measures the depth or the steepness of the graph and a function at a particular point on the graph. The derivative is called a slope, as it is the ratio of the change in the value of a specific function to a change in that independent variable's value.