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.
FAQs on Value of Log 4
1. What is the exact value of log 4 to the base 10?
The value of log 4 to the base 10, also known as the common logarithm, is approximately 0.60206. As an irrational number, its decimal representation continues infinitely without repeating. For most calculations as per the CBSE curriculum, using the four-digit value 0.6021 is standard and sufficient.
2. How is the value of log 4 calculated step-by-step?
To calculate the value of log 4, we use the properties of logarithms. The steps are as follows:
- First, express the number 4 as a power of a smaller, base number. In this case, 4 = 2².
- Substitute this into the logarithm expression, which gives you log(2²).
- Next, apply the power rule of logarithms, which states that log(aⁿ) = n * log(a). Applying this rule transforms the expression to 2 * log(2).
- Finally, use the known value of log 2 (which is approximately 0.3010). The calculation becomes 2 × 0.3010 = 0.6020.
3. Why is the value of log 4 simply double the value of log 2?
The value of log 4 is exactly double that of log 2 due to a fundamental property of logarithms called the Power Rule. Since 4 can be written exponentially as 2², the expression log(4) is identical to log(2²). The Power Rule allows us to move the exponent from inside the logarithm to become a multiplier in front, resulting in the expression 2 * log(2). This mathematical relationship directly shows that finding log 4 is equivalent to doubling the value of log 2.
4. What is the value of the natural logarithm of 4 (ln 4)?
The natural logarithm of 4, written as ln(4) or logₑ(4), uses the mathematical constant 'e' (approximately 2.718) as its base. The value of ln(4) is approximately 1.38629. This is different from the common log (base 10) and is a key concept in higher-level mathematics, especially in topics related to calculus and exponential growth for the 2025-26 syllabus.
5. How does the value of log 4 (base 10) differ from the value of log 4 with base 2?
The value of a logarithm is entirely dependent on its base, as the base defines the question being asked. The key difference is:
- log₁₀(4) ≈ 0.6021: This asks, "To what power must 10 be raised to get the number 4?"
- log₂(4) = 2: This asks, "To what power must 2 be raised to get the number 4?" The answer is exactly 2, because 2² = 4.
6. What are the key logarithmic properties needed to find the value of log 4?
To find the value of log 4, the most important property is the Power Rule, which states logₐ(mⁿ) = n * logₐ(m). This allows you to simplify log(2²) into 2*log(2). Alternatively, the Product Rule, which is log(a × b) = log(a) + log(b), can also be used. By this rule, log(4) = log(2 × 2) = log(2) + log(2), which also simplifies to 2*log(2).
7. Can the value of log 4 ever be negative?
No, the value of log 4, when using a standard base greater than 1 (like 10 or 'e'), cannot be negative. The properties of logarithms dictate that for any base b > 1:
- The logarithm is positive for numbers greater than 1 (e.g., log 4).
- The logarithm is zero for the number 1 (e.g., log 1 = 0).
- The logarithm is negative for numbers between 0 and 1 (e.g., log 0.4).
8. Where is the concept behind calculating log 4 applied in real life?
The principles used to calculate log 4 are fundamental to logarithmic scales, which are used to measure quantities that vary over a vast range. Key real-world applications include:
- Richter Scale: To measure earthquake intensity, where each whole number increase represents a tenfold increase in magnitude.
- Decibel (dB) Scale: To measure the loudness of sound.
- pH Scale in Chemistry: To determine the acidity or alkalinity of a solution based on hydrogen ion concentration.
- Finance: To calculate compound interest and model investment growth.
