Value of Log 4

Logarithms is 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 below.

Properties of Logarithmic Functions

• Any logarithmic function of ‘x’ can be represented as \[Log_{a} X = Y\]. This can also be depicted equivalently in the form of exponents as \[X = a^{Y}\].

\[Log_{a} X = Y \Rightarrow  X = a^{Y}\]

• 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. 

\[log_{b} (XY) = log_{b} X + log_{b} Y\]

• 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.

\[log_{b} (\frac{X}{Y}) = log_{b} X - log_{b} Y\]

• 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.

\[log_{b} (X^{Y}) = Y log_{b} X\]

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

\[ log_{b} X = 0\] 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. 

\[4 = 2^{2}\]

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

\[Log_{10} 4 =  Log_{10}2^{2}\]

Step 3:

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

\[Log_{10} 4 = 2 Log_{10}2\]

Step 4: 

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

\[Log_{a}X = Y \Rightarrow  X = a^{Y}\]

If \[Log_{10}2  = Y\], then it can be written in the form of exponents as \[2 = 10^{Y}\]

Step 5: 

Determine the value of Y which gives the value of \[Log_{10}2\].

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 \[log_{10}2\] is the equation represented in step 3 to obtain a log of 4 to the base 10.

\[Log_{10} 4 = 2 Log_{10}2\] = 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 \[log_{e}4\] 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. 

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. 

 \[log_{e}4\] = ln 4 = 0.60206 x 2.303

 \[log_{e}4\] = ln 4 = 1.386

With accurate and precise computations, the natural log 4 value is calculated up to 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. 

\[4 = 2^{2}\]

Step 2: 

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

\[Log_{2} 4 =  Log_{2}2^{2}\]

Step 3:

Using the power rule of logarithms, log 4 can be written as the product of 2 and \[log_{2}2\]

\[Log_{2} 4 = 2 Log_{2}2\]

Step 4: 

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

\[Log_{a}X = Y \Rightarrow  X = a^{Y}\]

If \[Log_{2}2 = Y\], then it can be written in the form of exponents as \[2 = 2^{Y}\]. 

Step 5: 

Determine the value of Y which gives the value of \[Log_{2}2\].

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

Step 6: 

Substitute the value of \[log_{2}2\] in the equation represented in step 3 to get the log 4 value to the base 2.

\[Log_{2} 4 = 2 Log_{2}2 = 2 \times 1 = 2\] 

Fun Facts:

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

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

The above-mentioned text introduced you to the value of log 4 and also the related properties. It also stated the step-by-step explanation. If you want more examples you can get them on the Vedantu website or the mobile app.