Value of log 3

Value of log⁡(3) is 0.4771 in base 10 (common logarithm) and log⁡(3) is 1.098612 in base e (natural logarithm) . A logarithm is a mathematical function that expresses the power to which a base must be raised to produce a given number or we can say it is a different way to represent the exponent.

👁 Value-of-Log-3

What is Logarithm?

A logarithm, denoted as “log,” is a mathematical function that represents the exponent to which a specified base must be raised to obtain a given number. In other words, it’s a way of expressing an exponent.

The logarithm of a number “x” to the base “b” is denoted as “log_b(x)” and is defined as:

log_b(x) = y

Value of log 3 base 10

Log base 10 is also called a common logarithm. The value of log 3 base 10 is given below:

log₁₀3 = log 3 ≈ 1.098612

Value of log 3 base e

Log base e is also called Natural Logarithm. The value of log 3 base e is given below:

log_e3 = ln 3 ≈ 0.4771

How to Calculate the Value of Log 3?

To calculate the value of log 3 there are two methods i.e.,

• Using Log Table
• Using Taylor Series

Discuss these methods as follows:

Using a Log Table

For any number without any decimal, it is very easy to find the value using a log table as we can see

Step 1: Find a Log Table with base 10.

Step 2: Locate the row corresponding to the mantissa (3).

Step 3: Identify the column for the characteristic (whole number part) which is 0 here.

Step 4: The intersection of row and column provides the logarithm value (log 3 ≈ 0.4771).

Read More about Log Table.

Using Taylor Series

To calculate the value of logarithm of 3 using Taylor series expansion, you can use the Taylor series expansion of the natural logarithm function. The natural logarithm of x can be represented as:

ln(x) = (x - 1) - (1/2)(x - 1)² + (1/3)(x - 1)³ - (1/4)(x - 1)⁴ + . . .

To calculate ln(3), you would plug in x = 3 into this series expansion.

ln(3) ≈ (3 - 1) - (1/2)(3 - 1)² + (1/3)(3 - 1)³ - (1/4)(3 - 1)⁴ + . . .

⇒ ln(3) ≈ 2 - 1 + 2/3 - 1/2

⇒ ln(3) ≈ 1 + 4/6 - 3/6 ≈ 1 + 1/6 ≈ 1.166666. . .

As we know log x = ln x/ln 10, and ln 10 = 2.3026

Thus, log 3 = ln 3/2.3026 ≈ 0.51

Note: The higher the number of terms taken from the series expansion, more accurate is the result.

Conclusion

Log 3 represents the power to which the base 10 must be raised to produce the number 3.It is a mathematical concept which is widely used in various fields such as engineering, mathematics and science. In this article we discussed about what is logarithm , the value of log 3 base 10 and base e. We also discussed how we can calculate the value of log 3 by using log table and using logarithmic formulaes.

Sample Problems on Value of Log 3

Problem 1: Find the value of log 3³.

Solution:

log ab = b × log a

log 33 = 3 × log 3

= 3× 0.4471

≈ 1.4313

Problem 2: find the value of x, 10^x = 3 ?

Solution:

10^x = 3

Taking log base 10 both sides, we get

log₁₀10^x = log₁₀3

⇒ x log₁₀10 = log₁₀3

⇒ x = log₁₀ 3 [log₁₀10 = 1]

⇒ x = log10 3 / log10 10

⇒ x = 0.4471/ 1

⇒ x = 0.4471