Value of Log 4

Introduction:

Logarithms are the inverse functions of exponential functions. Logarithm was first designed and used by John Napier. There are two types of logarithms namely natural logarithms and common logarithms. Natural logarithms are the logarithmic functions with the base value as ‘e’ a mathematical constant equal to 2.71828 whereas common logarithms are the logarithmic functions with a base value equal to 10. Logarithms are generally used in complex mathematical, scientific, and statistical computations. Logarithmic values of positive integer 4 (log of 4) calculated with base 10 and base ‘e’ are given as follows. 


Common Logarithm of 4 = Log4 = 0.60206

Natural Logarithm of 4 = ln 4 = 1.386294

Logarithm to the base 2 of 4 = Log 24 = 2


Properties of Logarithmic Functions:

  • Any logarithmic function of ‘x’ can be represented as LogaX = Y. This can also be depicted equivalently in the form of exponents as X = aY.


LogaX = Y => X = aY


  • The logarithm of a product of two numbers or variables is equal to the sum of the logarithmic values of individual numbers or variables. This rule is called the product rule. 


logb (XY) = logb X + logb Y


  • The logarithm of a quotient of two numbers or variables is given as the subtraction of the logarithmic value of the divisor from the logarithmic value of the dividend. This is called the quotient rule.

logb (X / Y) = logb X - logb Y


  • Logarithmic value of the power of a number or variable is given as the product of the value in power and the logarithmic value of the number or variable. This is called the power rule.

logb (XY) = Y logb X


  • Derivative of the logarithmic function of a constant value is equal to zero.

logb X = 0 if X is a mathematical constant



Calculating the value of log4 to the base 10:

Step 1:

4 is a perfect square number. It can be represented as 2 to the power 2. 


4 = 22 


Step 2: 

Apply logarithmic function to the base 10 on both sides of the above equation. Log of 4 to the base 10 is given as:

Log 104 = Log 1022


Step 3:

Using the power rule of logarithms, log4 can be written as the product of 2 and log 2.


Log 104 = 2 Log 102


Step 4: 

Calculate the value of log 2 to the base 10 using the inverse of logarithmic function or exponential function as follows:

LogaX = Y => X = aY


If Log102 = Y, then it can be written in the form of exponents as 2 = 10Y 


Step 5: 

Determine the value of Y which gives the value of Log 102.

There is no simple method to calculate the value of Y in the equation depicted in step 4. However, the value can be determined using a scientific calculator. A few complex mathematical calculations give the value of Y as ‘0.30103’. 


Step 6: 

Substitute the value of log 102 is the equation represented in step 3 to obtain log of 4 to the base 10.

Log 104 = 2 Log 102 = 2 x 0.30103 = 0.60206


Calculating the value of log 4 to the base ‘e’:

Natural logarithm of positive integer 4 is represented as log e4 or ln 4. The base of a natural logarithmic function is ‘e’, a Mathematical constant equal to 2.71828.

Common logarithmic value and natural logarithmic value of any number ‘X’ is related as shown below. 


Natural logarithmic value = Common logarithmic value x 2.303


As we have already calculated the log 4 value to the base 10, the natural logarithm of 4 can be calculated by multiplying it with the number 2.303. 

log e4 = ln 4 = 0.60206 x 2.303

log e4 = ln 4 = 1.386

With accurate and precise computations, the natural log 4 value is calculated upto six decimal places as 1.386294.

Calculating the value of log4 to the base 2:

Step 1:

4 is a perfect square number. It can be represented as 2 to the power 2. 


4 = 22 


Step 2: 

Apply logarithmic function to the base 2 on both sides of the above equation.


Log 24 = Log 222


Step 3:

Using the power rule of logarithms, log 4 can be written as the product of 2 and log 22.


Log 24 = 2 Log 22


Step 4: 

Calculate the value of log 2 to the base 2 using the inverse of logarithmic function or exponential function as follows:

LogaX = Y => X = aY

If Log 22 = Y, then it can be written in the form of exponents as 2 = 2Y 


Step 5: 

Determine the value of Y which gives the value of Log 22.

Since the bases are the same, the powers can be equated. Therefore Y = 1.


Step 6: 

Substitute the value of log 22 is the equation represented in step 3 to get the log 4 value to the base 2.


Log 24 = 2 Log 22 = 2 x 1 = 2


Example Problems:



Solution: 

64 can be written as 4 x 4 x 4 = 43 and 1/ 64 can be written as 4-3 

          = Log 24-3 

  = -3 Log 24           (Power rule of logarithms)

                                      = - 3 (2)           (Substitute the value of Log 24)

                                      = - 6



Solution: 

  =      Log 101024 - Log 1016 (Quotient rule)

= Log 1045 - Log 1042 (Representing nos in exponential form)

= 5 Log 104 - 2 Log 104 (Power rule)

= 5 (0.60206) - 2 (0.60206)     (Substitute Log 104 = 0.60206)

= 3 (0.60206) 

= 1.80618


Fun Facts:

  • The value of log 4 to the base 4 is equal to unity.

  • Antilogarithm of logarithmic value of 4 is equal to 4.

FAQ (Frequently Asked Questions)

1. How does the logarithmic value of 4 vary with the base number?

Logarithmic function can mathematically be defined as the inverse of an exponential function. Logarithmic function a number X can be written as LogaX = Y. The same equation can be written in exponential form as X = aY. Here ‘a’ is the base of the logarithmic function. It may take the value of any positive integer. The value of log 4 can be calculated using the base as ‘10’, ‘e’, and ‘2’. The logarithmic function of 4 to the base 10 is equal to 0.60206. The natural logarithmic value of 4 is 1.386294. Logarithmic value of 4 to the base 2 = 2 and the logarithmic value of 4 to the base 4 is equal to 1. So, from a keen observation of these values, we can infer that the logarithmic value of a number decreases with the increase in the value of the base.

2. Where are logarithmic functions used?

Logarithmic functions are used in various scientific and mathematical calculations. A few notable calculations in which logarithms are used include:

  • Measuring earthquake intensity using Ritcher scale

  • Estimating the value of pH

  • Calculating the intensity of sound in terms of decibels

  • Representing large mathematical numbers

  • Calculations involving complex numerical values

  • The difference in values of natural logarithms and common logarithms of any number is used in several comparisons.